answer：First, I need to determine how many Nintendo games Kelly will have left after giving away 138 games from her total of 257 games.To find this, I will subtract the number of games she gives away from her total number of games.So, I will perform the calculation: 257 minus 138.This will give me the number of games Kelly has remaining.

answer：Alright, so I have this geometry problem here. It says: Given a circle ( k ) and a line ( e ), with points ( P ) on ( k ) and ( Q ) on ( e ), I need to construct two circles of equal radius that touch each other. One of these circles should touch the given circle ( k ) at point ( P ), and the other should touch the line ( e ) at point ( Q ).Hmm, okay. Let me try to visualize this. There's a circle ( k ) and a straight line ( e ). Point ( P ) is somewhere on the circumference of ( k ), and point ( Q ) is somewhere on the line ( e ). I need to create two new circles. Both of these circles should have the same radius. They should also touch each other, meaning they are tangent to each other. One of these circles must be tangent to the original circle ( k ) exactly at point ( P ), and the other must be tangent to the line ( e ) exactly at point ( Q ).Alright, so first, I need to think about what it means for two circles to touch each other. If two circles touch each other, they can either touch externally or internally. Since the problem doesn't specify, I might assume they touch externally, meaning they each lie outside the other and touch at exactly one point.Next, I need to consider the conditions for a circle to be tangent to another circle or a line. For a circle to be tangent to another circle at a specific point, their centers and that point must lie on the same straight line. Similarly, for a circle to be tangent to a line at a specific point, the center of the circle must lie along the perpendicular to the line at that point.So, for the circle that's tangent to circle ( k ) at ( P ), its center must lie along the line that is the radius of ( k ) at point ( P ). That is, if I draw a line from the center of ( k ) through ( P ), the center of the new circle must lie somewhere along this line.Similarly, for the circle that's tangent to line ( e ) at ( Q ), its center must lie along the perpendicular to ( e ) at point ( Q ). So, if I draw a line perpendicular to ( e ) at ( Q ), the center of the second new circle must lie somewhere along this perpendicular.Since both new circles have the same radius, let's call this radius ( r ). The distance between the centers of these two new circles must be equal to twice the radius, ( 2r ), because they are tangent to each other externally.Let me try to summarize the steps I need to take:1. Identify the center of the original circle ( k ). Let's call this center ( O ).2. Draw the radius ( OP ) of circle ( k ). The center of the first new circle, let's call it ( C_1 ), must lie somewhere along this radius.3. Draw the perpendicular to line ( e ) at point ( Q ). The center of the second new circle, ( C_2 ), must lie somewhere along this perpendicular.4. The distance between ( C_1 ) and ( C_2 ) must be ( 2r ), and both ( C_1 ) and ( C_2 ) must be at a distance ( r ) from their respective tangent points ( P ) and ( Q ).Wait, actually, the distance from ( C_1 ) to ( P ) should be ( r ), and the distance from ( C_2 ) to ( Q ) should also be ( r ). But since ( C_1 ) is along ( OP ), and ( C_2 ) is along the perpendicular at ( Q ), I need to find points ( C_1 ) and ( C_2 ) such that:- ( C_1P = r )- ( C_2Q = r )- ( C_1C_2 = 2r )This seems like a system of equations. Maybe I can set up coordinate axes to solve this algebraically.Let's assign coordinates to make this easier. Let me place point ( P ) at the origin for simplicity. Wait, but ( P ) is on circle ( k ), so maybe it's better to place the center ( O ) of circle ( k ) at the origin. Then, point ( P ) can be at some coordinate, say ( (a, 0) ) if I align the radius ( OP ) along the x-axis.Similarly, line ( e ) can be represented as a line in the coordinate system. Let's say line ( e ) is horizontal for simplicity, so its equation can be ( y = c ), and point ( Q ) is at ( (b, c) ).Wait, but if I place ( O ) at the origin, and ( P ) at ( (a, 0) ), then the radius of circle ( k ) is ( a ). So, circle ( k ) has equation ( x^2 + y^2 = a^2 ).Line ( e ) is ( y = c ), and point ( Q ) is ( (b, c) ).Now, the center ( C_1 ) of the first new circle must lie along the radius ( OP ), which is the x-axis. So, ( C_1 ) has coordinates ( (h, 0) ) for some ( h ). The distance from ( C_1 ) to ( P ) is ( |h - a| = r ). So, ( h = a pm r ).Similarly, the center ( C_2 ) of the second new circle must lie along the perpendicular to ( e ) at ( Q ). Since ( e ) is horizontal, the perpendicular is vertical. So, ( C_2 ) has coordinates ( (b, c + k) ) for some ( k ). The distance from ( C_2 ) to ( Q ) is ( |k| = r ). So, ( k = pm r ).Now, the distance between ( C_1 ) and ( C_2 ) must be ( 2r ). So, the distance between ( (h, 0) ) and ( (b, c + k) ) is:[sqrt{(h - b)^2 + (0 - (c + k))^2} = 2r]Substituting ( h = a pm r ) and ( k = pm r ), we get:[sqrt{(a pm r - b)^2 + ( - (c pm r))^2} = 2r]Squaring both sides:[(a pm r - b)^2 + (c pm r)^2 = 4r^2]This equation can be solved for ( r ). Let's expand it:[(a - b pm r)^2 + (c pm r)^2 = 4r^2]Expanding each term:[(a - b)^2 pm 2(a - b)r + r^2 + c^2 pm 2cr + r^2 = 4r^2]Combine like terms:[(a - b)^2 + c^2 pm 2(a - b)r pm 2cr + 2r^2 = 4r^2]Bring all terms to one side:[(a - b)^2 + c^2 pm 2(a - b)r pm 2cr + 2r^2 - 4r^2 = 0]Simplify:[(a - b)^2 + c^2 pm 2(a - b)r pm 2cr - 2r^2 = 0]This is a quadratic equation in terms of ( r ). Let's write it as:[-2r^2 pm 2(a - b + c)r + (a - b)^2 + c^2 = 0]Multiply both sides by -1 to make it more standard:[2r^2 mp 2(a - b + c)r - (a - b)^2 - c^2 = 0]Now, this is a quadratic equation of the form ( Ar^2 + Br + C = 0 ), where:- ( A = 2 )- ( B = mp 2(a - b + c) )- ( C = - (a - b)^2 - c^2 )We can solve for ( r ) using the quadratic formula:[r = frac{-B pm sqrt{B^2 - 4AC}}{2A}]Plugging in the values:[r = frac{pm 2(a - b + c) pm sqrt{[ mp 2(a - b + c) ]^2 - 4 cdot 2 cdot [ - (a - b)^2 - c^2 ]}}{2 cdot 2}]Simplify inside the square root:First, compute ( B^2 ):[[ mp 2(a - b + c) ]^2 = 4(a - b + c)^2]Then compute ( 4AC ):[4 cdot 2 cdot [ - (a - b)^2 - c^2 ] = 8[ - (a - b)^2 - c^2 ] = -8(a - b)^2 - 8c^2]So, the discriminant becomes:[4(a - b + c)^2 - (-8(a - b)^2 - 8c^2) = 4(a - b + c)^2 + 8(a - b)^2 + 8c^2]Let me expand ( (a - b + c)^2 ):[(a - b + c)^2 = (a - b)^2 + 2(a - b)c + c^2]So, substituting back:[4[(a - b)^2 + 2(a - b)c + c^2] + 8(a - b)^2 + 8c^2][= 4(a - b)^2 + 8(a - b)c + 4c^2 + 8(a - b)^2 + 8c^2][= (4 + 8)(a - b)^2 + 8(a - b)c + (4 + 8)c^2][= 12(a - b)^2 + 8(a - b)c + 12c^2]So, the discriminant is:[12(a - b)^2 + 8(a - b)c + 12c^2]Therefore, the expression for ( r ) becomes:[r = frac{pm 2(a - b + c) pm sqrt{12(a - b)^2 + 8(a - b)c + 12c^2}}{4}]Simplify the square root:Factor out 4 from the square root:[sqrt{4[3(a - b)^2 + 2(a - b)c + 3c^2]} = 2sqrt{3(a - b)^2 + 2(a - b)c + 3c^2}]So, substituting back:[r = frac{pm 2(a - b + c) pm 2sqrt{3(a - b)^2 + 2(a - b)c + 3c^2}}{4}][= frac{pm (a - b + c) pm sqrt{3(a - b)^2 + 2(a - b)c + 3c^2}}{2}]Since radius cannot be negative, we can discard the negative solutions. Therefore, we have:[r = frac{(a - b + c) + sqrt{3(a - b)^2 + 2(a - b)c + 3c^2}}{2}]or[r = frac{(a - b + c) - sqrt{3(a - b)^2 + 2(a - b)c + 3c^2}}{2}]But we need to check if these solutions are positive. The second solution might result in a negative radius, which isn't possible, so we might only consider the first solution.Wait, but this seems complicated. Maybe I made a mistake in setting up the coordinates. Let me try a different approach.Instead of assigning coordinates immediately, perhaps I can use geometric constructions.First, I know that the center ( C_1 ) of the first circle lies along the radius ( OP ) of circle ( k ). Similarly, the center ( C_2 ) of the second circle lies along the perpendicular to ( e ) at ( Q ).Let me denote the radius of the two new circles as ( r ). Then, the distance from ( C_1 ) to ( P ) is ( r ), and the distance from ( C_2 ) to ( Q ) is ( r ).Since the two new circles are tangent to each other, the distance between ( C_1 ) and ( C_2 ) must be ( 2r ).So, if I can find points ( C_1 ) along ( OP ) and ( C_2 ) along the perpendicular at ( Q ) such that ( C_1C_2 = 2r ), and ( C_1P = C_2Q = r ), then I can construct the circles.This seems like a problem that can be solved using similar triangles or perhaps by constructing a system of circles.Let me think about the positions of ( C_1 ) and ( C_2 ). Since ( C_1 ) is along ( OP ), and ( C_2 ) is along the perpendicular at ( Q ), the line connecting ( C_1 ) and ( C_2 ) must have length ( 2r ).Perhaps I can construct a circle centered at ( P ) with radius ( r ), and another circle centered at ( Q ) with radius ( r ), and find their intersection points. But wait, that might not directly help because ( C_1 ) and ( C_2 ) are not necessarily related to ( P ) and ( Q ) in that way.Alternatively, maybe I can use the method of loci. The set of all possible centers ( C_1 ) such that ( C_1P = r ) is a circle of radius ( r ) centered at ( P ). Similarly, the set of all possible centers ( C_2 ) such that ( C_2Q = r ) is a circle of radius ( r ) centered at ( Q ). The intersection points of these two circles would give possible centers ( C_1 ) and ( C_2 ), but they also need to lie along their respective lines ( OP ) and the perpendicular at ( Q ).This seems a bit convoluted. Maybe I need to use inversion or some other geometric transformation, but I'm not sure.Wait, perhaps I can use the concept of homothety. If I consider the two new circles to be tangent to each other, there might be a homothety that maps one circle to the other, centered at their point of tangency.But I'm not sure how to apply that here. Maybe I should go back to the coordinate system approach.Let me try again with coordinates, but this time, I'll place point ( P ) at ( (0, 0) ) for simplicity. Then, the center ( O ) of circle ( k ) would be at ( (d, 0) ), where ( d ) is the radius of circle ( k ). So, circle ( k ) has equation ( (x - d)^2 + y^2 = d^2 ).Point ( Q ) is on line ( e ). Let me assume line ( e ) is the x-axis for simplicity, so ( e ) is ( y = 0 ), and point ( Q ) is at ( (q, 0) ).Wait, but if ( e ) is the x-axis and ( P ) is also on the x-axis, that might complicate things because both circles would be tangent to the x-axis. Maybe I should choose a different line for ( e ). Let me make line ( e ) a horizontal line above the x-axis, say ( y = h ), and point ( Q ) is at ( (q, h) ).So, circle ( k ) has center ( (d, 0) ) and radius ( d ), and line ( e ) is ( y = h ) with point ( Q ) at ( (q, h) ).Now, the center ( C_1 ) of the first new circle lies along the radius ( OP ). Since ( P ) is at ( (0, 0) ) and ( O ) is at ( (d, 0) ), the radius ( OP ) is along the x-axis. Therefore, ( C_1 ) must lie somewhere along the x-axis. Let's denote ( C_1 ) as ( (c, 0) ).The distance from ( C_1 ) to ( P ) is ( |c - 0| = |c| = r ). So, ( c = pm r ). But since ( C_1 ) is outside circle ( k ), and ( k ) has radius ( d ), ( C_1 ) must be at ( (d + r, 0) ) or ( (d - r, 0) ). Wait, but ( P ) is at ( (0, 0) ), so if ( C_1 ) is along ( OP ), which is from ( (d, 0) ) to ( (0, 0) ), then ( C_1 ) must be between ( O ) and ( P ) or beyond ( P ).If ( C_1 ) is between ( O ) and ( P ), then ( c = d - r ). If it's beyond ( P ), then ( c = -r ). But since ( P ) is on circle ( k ), and we want the new circle to touch ( k ) at ( P ), ( C_1 ) must be outside circle ( k ). Therefore, ( C_1 ) must be at ( (d + r, 0) ).Wait, no. If ( C_1 ) is the center of the new circle tangent to ( k ) at ( P ), then ( C_1 ) must lie along the line ( OP ), but on the opposite side of ( P ) relative to ( O ). So, if ( O ) is at ( (d, 0) ) and ( P ) is at ( (0, 0) ), then ( C_1 ) must be at ( (-r, 0) ), because the distance from ( C_1 ) to ( P ) is ( r ).Wait, that makes more sense. Because if ( C_1 ) is at ( (-r, 0) ), then the distance from ( C_1 ) to ( P ) is ( r ), and the new circle centered at ( (-r, 0) ) with radius ( r ) will touch circle ( k ) at ( P ).Similarly, the center ( C_2 ) of the second new circle must lie along the perpendicular to ( e ) at ( Q ). Since ( e ) is ( y = h ), the perpendicular is vertical. So, ( C_2 ) must lie along the vertical line passing through ( Q ), which is ( x = q ). Let's denote ( C_2 ) as ( (q, k) ).The distance from ( C_2 ) to ( Q ) is ( |k - h| = r ). So, ( k = h pm r ). Since the new circle must lie above or below line ( e ), depending on the configuration. If we assume it's above, then ( k = h + r ). If below, ( k = h - r ).Now, the distance between ( C_1 ) and ( C_2 ) must be ( 2r ). So, the distance between ( (-r, 0) ) and ( (q, k) ) is:[sqrt{(q + r)^2 + (k - 0)^2} = 2r]Substituting ( k = h pm r ):[sqrt{(q + r)^2 + (h pm r)^2} = 2r]Squaring both sides:[(q + r)^2 + (h pm r)^2 = 4r^2]Expanding:[q^2 + 2qr + r^2 + h^2 pm 2hr + r^2 = 4r^2]Combine like terms:[q^2 + h^2 + 2qr pm 2hr + 2r^2 = 4r^2]Bring all terms to one side:[q^2 + h^2 + 2qr pm 2hr + 2r^2 - 4r^2 = 0][q^2 + h^2 + 2qr pm 2hr - 2r^2 = 0]This is a quadratic equation in terms of ( r ):[-2r^2 + (2q pm 2h)r + (q^2 + h^2) = 0]Multiply both sides by -1:[2r^2 - (2q pm 2h)r - (q^2 + h^2) = 0]Divide both sides by 2:[r^2 - (q pm h)r - frac{q^2 + h^2}{2} = 0]Now, solve for ( r ) using the quadratic formula:[r = frac{(q pm h) pm sqrt{(q pm h)^2 + 2(q^2 + h^2)}}{2}]Let me compute the discriminant:[D = (q pm h)^2 + 2(q^2 + h^2)][= q^2 pm 2qh + h^2 + 2q^2 + 2h^2][= 3q^2 pm 2qh + 3h^2]So,[r = frac{(q pm h) pm sqrt{3q^2 pm 2qh + 3h^2}}{2}]Again, we need ( r ) to be positive, so we take the positive roots.This gives us possible solutions for ( r ). Depending on the signs, we might have different configurations.But this seems quite involved. Maybe there's a simpler geometric construction.Let me think about the problem again. We have two circles of equal radius ( r ), one tangent to circle ( k ) at ( P ), and the other tangent to line ( e ) at ( Q ). They are also tangent to each other.If I can find the centers ( C_1 ) and ( C_2 ) such that:1. ( C_1 ) lies along the radius ( OP ) of circle ( k ).2. ( C_2 ) lies along the perpendicular to ( e ) at ( Q ).3. The distance between ( C_1 ) and ( C_2 ) is ( 2r ).4. The distance from ( C_1 ) to ( P ) is ( r ).5. The distance from ( C_2 ) to ( Q ) is ( r ).Perhaps I can use the method of loci again. The locus of ( C_1 ) is along ( OP ), and the locus of ( C_2 ) is along the perpendicular at ( Q ). The intersection of the circles centered at ( C_1 ) and ( C_2 ) with radius ( r ) should give the point of tangency.Wait, maybe I can construct the circles step by step.First, draw circle ( k ) with center ( O ) and point ( P ) on its circumference. Draw line ( e ) with point ( Q ) on it.Next, draw the radius ( OP ). The center ( C_1 ) of the first new circle must lie along this radius. Similarly, draw the perpendicular to ( e ) at ( Q ); the center ( C_2 ) of the second new circle must lie along this perpendicular.Now, the distance between ( C_1 ) and ( C_2 ) must be ( 2r ), and both ( C_1 ) and ( C_2 ) must be at a distance ( r ) from ( P ) and ( Q ), respectively.Perhaps I can use similar triangles or the Pythagorean theorem to relate these distances.Let me denote the distance between ( O ) and ( Q ) as ( d ). Then, using the Pythagorean theorem, I can relate ( r ), ( d ), and the other distances.Wait, but I don't know the exact positions of ( O ) and ( Q ). Maybe I need to express everything in terms of coordinates again.Alternatively, perhaps I can use the concept of inversion. Inversion can sometimes simplify problems involving tangent circles. But I'm not very familiar with inversion techniques, so I might need to look that up.Wait, maybe I can use the method of coordinates again, but this time, I'll assign specific coordinates to make the math easier.Let me place the center ( O ) of circle ( k ) at ( (0, 0) ), and let the radius of ( k ) be ( R ). So, point ( P ) is at ( (R, 0) ).Line ( e ) can be represented as ( y = c ), and point ( Q ) is at ( (q, c) ).Now, the center ( C_1 ) of the first new circle lies along the radius ( OP ), which is the x-axis. So, ( C_1 ) is at ( (R + r, 0) ) or ( (R - r, 0) ). But since the new circle must be outside circle ( k ), ( C_1 ) must be at ( (R + r, 0) ).The center ( C_2 ) of the second new circle lies along the perpendicular to ( e ) at ( Q ). Since ( e ) is horizontal, the perpendicular is vertical. So, ( C_2 ) is at ( (q, c + r) ) or ( (q, c - r) ). Assuming it's above ( e ), ( C_2 ) is at ( (q, c + r) ).Now, the distance between ( C_1 ) and ( C_2 ) must be ( 2r ):[sqrt{(q - (R + r))^2 + (c + r - 0)^2} = 2r]Squaring both sides:[(q - R - r)^2 + (c + r)^2 = 4r^2]Expanding:[(q - R)^2 - 2(q - R)r + r^2 + c^2 + 2cr + r^2 = 4r^2]Combine like terms:[(q - R)^2 + c^2 - 2(q - R)r + 2cr + 2r^2 = 4r^2]Bring all terms to one side:[(q - R)^2 + c^2 - 2(q - R)r + 2cr + 2r^2 - 4r^2 = 0][(q - R)^2 + c^2 - 2(q - R)r + 2cr - 2r^2 = 0]This is a quadratic equation in ( r ):[-2r^2 + [ -2(q - R) + 2c ]r + (q - R)^2 + c^2 = 0]Multiply by -1:[2r^2 + [2(q - R) - 2c]r - (q - R)^2 - c^2 = 0]Divide by 2:[r^2 + [ (q - R) - c ]r - frac{(q - R)^2 + c^2}{2} = 0]Now, solve for ( r ):[r = frac{ -[ (q - R) - c ] pm sqrt{ [ (q - R) - c ]^2 + 2[(q - R)^2 + c^2] } }{2}]Simplify the discriminant:[D = [ (q - R) - c ]^2 + 2[(q - R)^2 + c^2]][= (q - R)^2 - 2(q - R)c + c^2 + 2(q - R)^2 + 2c^2][= 3(q - R)^2 - 2(q - R)c + 3c^2]So,[r = frac{ -[ (q - R) - c ] pm sqrt{3(q - R)^2 - 2(q - R)c + 3c^2} }{2}]Since ( r ) must be positive, we take the positive root:[r = frac{ -[ (q - R) - c ] + sqrt{3(q - R)^2 - 2(q - R)c + 3c^2} }{2}]This gives us the value of ( r ). Once ( r ) is determined, we can find the centers ( C_1 ) and ( C_2 ):- ( C_1 ) is at ( (R + r, 0) )- ( C_2 ) is at ( (q, c + r) )Then, we can draw the two circles centered at ( C_1 ) and ( C_2 ) with radius ( r ), ensuring they touch each other and are tangent to ( k ) and ( e ) at ( P ) and ( Q ), respectively.However, this algebraic approach, while giving an exact solution, might not be the most straightforward geometric construction. Perhaps there's a more elegant way using classical geometric constructions like compass and straightedge.Let me think about how to construct this without coordinates.1. Start by drawing circle ( k ) with center ( O ) and point ( P ) on its circumference.2. Draw line ( e ) with point ( Q ) on it.3. Draw the radius ( OP ) of circle ( k ).4. Draw the perpendicular to line ( e ) at point ( Q ).5. The centers ( C_1 ) and ( C_2 ) of the new circles must lie along ( OP ) and the perpendicular at ( Q ), respectively.6. The distance between ( C_1 ) and ( C_2 ) must be ( 2r ), and both must be at a distance ( r ) from ( P ) and ( Q ).Perhaps I can use the following steps:a. From point ( P ), draw a circle with radius ( r ); this will help locate ( C_1 ).b. From point ( Q ), draw a circle with radius ( r ); this will help locate ( C_2 ).c. The intersection of these two circles will give possible points for ( C_1 ) and ( C_2 ), but they must also lie along ( OP ) and the perpendicular at ( Q ).Wait, but since ( C_1 ) and ( C_2 ) are constrained to lie along specific lines, maybe I can use those lines to find their positions.Let me try:1. Draw circle ( k ) with center ( O ) and point ( P ) on its circumference.2. Draw line ( e ) with point ( Q ) on it.3. Draw the radius ( OP ).4. Draw the perpendicular to ( e ) at ( Q ).5. From ( P ), mark a point ( C_1 ) along ( OP ) such that ( PC_1 = r ).6. From ( Q ), mark a point ( C_2 ) along the perpendicular such that ( QC_2 = r ).7. Ensure that the distance ( C_1C_2 = 2r ).But how do I ensure that ( C_1C_2 = 2r )? This seems like the crux of the problem.Perhaps I can use the following method:- Construct a circle centered at ( P ) with radius ( r ); this will intersect ( OP ) at ( C_1 ).- Construct a circle centered at ( Q ) with radius ( r ); this will intersect the perpendicular at ( Q ) at ( C_2 ).- The intersection of these two circles will give the possible positions for ( C_1 ) and ( C_2 ), but they must also satisfy ( C_1C_2 = 2r ).Alternatively, since ( C_1 ) and ( C_2 ) are both at distance ( r ) from ( P ) and ( Q ), respectively, and at distance ( 2r ) from each other, we can consider the triangle ( PC_1C_2 ) with sides ( r ), ( 2r ), and the distance between ( P ) and ( Q ).Wait, the distance between ( P ) and ( Q ) is fixed, so maybe we can use the Law of Cosines.Let me denote the distance between ( P ) and ( Q ) as ( d ). Then, in triangle ( PC_1C_2 ):[d^2 = r^2 + (2r)^2 - 2 cdot r cdot 2r cdot cos theta][d^2 = r^2 + 4r^2 - 4r^2 cos theta][d^2 = 5r^2 - 4r^2 cos theta]But I don't know the angle ( theta ), so this might not help directly.Alternatively, since ( C_1 ) lies along ( OP ) and ( C_2 ) lies along the perpendicular at ( Q ), the line ( C_1C_2 ) forms a right triangle with ( OP ) and the perpendicular.Wait, if I consider the coordinates again, with ( O ) at ( (0, 0) ), ( P ) at ( (R, 0) ), ( Q ) at ( (q, c) ), ( C_1 ) at ( (R + r, 0) ), and ( C_2 ) at ( (q, c + r) ), then the distance ( C_1C_2 ) is:[sqrt{(q - R - r)^2 + (c + r)^2} = 2r]Which is the same equation as before. So, solving for ( r ) gives the required radius.Once ( r ) is found, constructing the circles is straightforward:1. Draw circle centered at ( C_1 ) with radius ( r ); it will touch circle ( k ) at ( P ).2. Draw circle centered at ( C_2 ) with radius ( r ); it will touch line ( e ) at ( Q ).3. Ensure that these two circles are tangent to each other.But since this requires solving a quadratic equation, it might not be possible with just compass and straightedge unless the discriminant is a perfect square.Alternatively, perhaps there's a geometric construction that avoids solving the quadratic equation.Let me think about the homothety center. If the two new circles are tangent to each other, there is a homothety that maps one to the other, centered at their point of tangency.Let me denote the point of tangency as ( T ). Then, the homothety centered at ( T ) maps ( C_1 ) to ( C_2 ) and scales by a factor of -1 (since they are externally tangent).But I'm not sure how to use this to construct ( C_1 ) and ( C_2 ).Wait, maybe I can use the method of similar triangles.Consider the triangle formed by ( O ), ( C_1 ), and ( C_2 ). Since ( C_1 ) is along ( OP ) and ( C_2 ) is along the perpendicular at ( Q ), the triangle ( OC_1C_2 ) might have some similar properties.Alternatively, perhaps I can construct a line from ( Q ) parallel to ( OP ) and find the intersection with the perpendicular, but I'm not sure.This problem is proving to be quite challenging. Maybe I need to look for a different approach or refer to some geometric theorems that can help.Wait, perhaps I can use the concept of the radical axis. The radical axis of two circles is the locus of points with equal power concerning both circles. But I'm not sure how that applies here.Alternatively, maybe I can use the method of inversion with respect to point ( P ) or ( Q ). Inversion can sometimes transform circles into lines or other circles, making the problem easier.But I'm not very familiar with inversion techniques, so I might need to research that.Alternatively, perhaps I can use the method of coordinates again, but this time, I'll assume specific values for ( R ), ( q ), and ( c ) to simplify the calculations.Let me assume ( R = 1 ), ( q = 0 ), and ( c = 1 ). So, circle ( k ) has center ( (0, 0) ) and radius 1, point ( P ) is at ( (1, 0) ), line ( e ) is ( y = 1 ), and point ( Q ) is at ( (0, 1) ).Now, the center ( C_1 ) is at ( (1 + r, 0) ), and the center ( C_2 ) is at ( (0, 1 + r) ).The distance between ( C_1 ) and ( C_2 ) is:[sqrt{(1 + r - 0)^2 + (0 - (1 + r))^2} = sqrt{(1 + r)^2 + (1 + r)^2} = sqrt{2(1 + r)^2} = sqrt{2}(1 + r)]This distance must equal ( 2r ):[sqrt{2}(1 + r) = 2r][sqrt{2} + sqrt{2}r = 2r][sqrt{2} = 2r - sqrt{2}r][sqrt{2} = r(2 - sqrt{2})][r = frac{sqrt{2}}{2 - sqrt{2}} = frac{sqrt{2}(2 + sqrt{2})}{(2 - sqrt{2})(2 + sqrt{2})} = frac{2sqrt{2} + 2}{4 - 2} = frac{2sqrt{2} + 2}{2} = sqrt{2} + 1]So, ( r = sqrt{2} + 1 ).Therefore, the centers are:- ( C_1 ) at ( (1 + sqrt{2} + 1, 0) = (sqrt{2} + 2, 0) )- ( C_2 ) at ( (0, 1 + sqrt{2} + 1) = (0, sqrt{2} + 2) )Wait, that can't be right because ( C_1 ) should be at ( (1 + r, 0) ), which with ( r = sqrt{2} + 1 ) would be ( (1 + sqrt{2} + 1, 0) = (sqrt{2} + 2, 0) ). Similarly, ( C_2 ) is at ( (0, 1 + r) = (0, 1 + sqrt{2} + 1) = (0, sqrt{2} + 2) ).But let's check the distance between ( C_1 ) and ( C_2 ):[sqrt{(sqrt{2} + 2 - 0)^2 + (0 - (sqrt{2} + 2))^2} = sqrt{(sqrt{2} + 2)^2 + (sqrt{2} + 2)^2} = sqrt{2(sqrt{2} + 2)^2} = (sqrt{2} + 2)sqrt{2} = 2 + 2sqrt{2}]But ( 2r = 2(sqrt{2} + 1) = 2sqrt{2} + 2 ), which matches. So, it works.Therefore, in this specific case, the radius ( r = sqrt{2} + 1 ), and the centers are at ( (sqrt{2} + 2, 0) ) and ( (0, sqrt{2} + 2) ).This suggests that the general solution involves solving a quadratic equation for ( r ), as we did earlier, and then constructing the centers accordingly.So, to summarize the construction steps:1. Given circle ( k ) with center ( O ) and point ( P ) on its circumference, and line ( e ) with point ( Q ) on it.2. Draw the radius ( OP ) of circle ( k ).3. Draw the perpendicular to line ( e ) at point ( Q ).4. Let ( R ) be the radius of circle ( k ), and let ( d ) be the distance between ( O ) and ( Q ).5. Using the coordinates or geometric relationships, set up the equation for ( r ) based on the distances and solve for ( r ).6. Once ( r ) is determined, mark the centers ( C_1 ) along ( OP ) at a distance ( r ) from ( P ), and ( C_2 ) along the perpendicular at ( Q ) at a distance ( r ) from ( Q ).7. Draw the two circles centered at ( C_1 ) and ( C_2 ) with radius ( r ); they will touch each other and be tangent to ( k ) and ( e ) at ( P ) and ( Q ), respectively.This construction works, but it relies on solving a quadratic equation to find ( r ). In a purely geometric construction without algebra, this might be challenging, but with the help of coordinates and algebra, it becomes feasible.Another approach could be to use the method of similar triangles or geometric transformations, but I think the coordinate method, while algebraic, provides a clear path to the solution.In conclusion, the key steps are:- Identify the constraints on the centers ( C_1 ) and ( C_2 ).- Set up the distance equations based on the given conditions.- Solve for ( r ) using algebraic methods.- Construct the circles based on the determined radius and centers.This ensures that the two new circles have equal radii, touch each other, and are tangent to the given circle and line at the specified points.

answer：First, I need to determine the total number of people in the group. There are 2 adults and 5 children, so the total number of people is 2 + 5 = 7.Next, I know that each meal costs 3. To find the total cost for all the meals, I multiply the number of people by the cost per meal: 7 people * 3 per meal = 21.Therefore, the total bill for the group is 21.

answer：First, I need to find the prime factorization of 210. I know that 210 can be broken down into its prime factors: 2, 3, 5, and 7.Since I'm looking for even divisors, each divisor must include the factor of 2. This means I can represent each even divisor as 2 multiplied by combinations of the other prime factors.Next, I'll consider all possible combinations of the remaining prime factors (3, 5, and 7) with their respective exponents of 0 or 1. This gives me the following even divisors:- 2- 2 × 3 = 6- 2 × 5 = 10- 2 × 7 = 14- 2 × 3 × 5 = 30- 2 × 3 × 7 = 42- 2 × 5 × 7 = 70- 2 × 3 × 5 × 7 = 210Now, I'll add up all these even divisors to find their total sum:2 + 6 + 10 + 14 + 30 + 42 + 70 + 210 = 384Therefore, the sum of all the even divisors of 210 is 384.

answer：Okay, so I have this problem where there are 6 seats in a row, and I need to seat 3 people. The condition is that exactly two empty seats are next to each other. I need to find out how many different seating arrangements satisfy this condition. The options given are A: 36, B: 48, C: 72, D: 96.First, I need to understand what exactly the problem is asking. We have 6 seats, and we're seating 3 people, which means there will be 3 empty seats. The key here is that exactly two of these empty seats must be adjacent, and the third empty seat must not be adjacent to these two. So, we can't have all three empty seats together, and we also can't have all three empty seats separated by at least one occupied seat.To approach this, I think I should consider the different ways the empty seats can be arranged such that exactly two are adjacent. Maybe I can model this by treating the two adjacent empty seats as a single unit and then see where this unit can be placed along with the third empty seat.Let me try to visualize the seats. There are 6 seats in a row: 1, 2, 3, 4, 5, 6.If I consider two empty seats as a unit, say seats 1 and 2, then the third empty seat can't be adjacent to this unit. So, the third empty seat can't be seat 3. It has to be either seat 4, 5, or 6. Similarly, if the two empty seats are in positions 2 and 3, the third empty seat can't be 1 or 4, so it has to be 5 or 6. Wait, but in this case, if the two empty seats are in the middle, the third empty seat has fewer options.Hmm, maybe I should think about how many ways I can place the two adjacent empty seats and then place the third empty seat such that it's not adjacent to the first two.Let's count the number of ways to place the two adjacent empty seats. In a row of 6 seats, the number of ways to choose two adjacent seats is 5: (1,2), (2,3), (3,4), (4,5), (5,6).For each of these, I need to place the third empty seat such that it's not adjacent to the two already chosen.Let's take each case one by one.1. Two empty seats at (1,2): The third empty seat can't be at 3. So, it can be at 4, 5, or 6. That's 3 choices.2. Two empty seats at (2,3): The third empty seat can't be at 1 or 4. So, it can be at 5 or 6. That's 2 choices.3. Two empty seats at (3,4): The third empty seat can't be at 2 or 5. So, it can be at 1 or 6. That's 2 choices.4. Two empty seats at (4,5): The third empty seat can't be at 3 or 6. So, it can be at 1 or 2. That's 2 choices.5. Two empty seats at (5,6): The third empty seat can't be at 4. So, it can be at 1, 2, or 3. That's 3 choices.So, adding these up: 3 + 2 + 2 + 2 + 3 = 12 ways to place the empty seats such that exactly two are adjacent.Now, for each of these 12 arrangements of empty seats, we need to seat 3 people in the remaining 3 seats. Since the people are distinguishable, the number of ways to arrange them is 3! = 6.Therefore, the total number of seating arrangements is 12 * 6 = 72.Wait, but let me double-check. Is there a possibility that some arrangements are being counted more than once? For example, if the two empty seats are at (1,2) and the third is at 4, is that different from two empty seats at (4,5) and the third at 1? Yes, because the positions are different.Also, I need to make sure that in all these cases, the third empty seat is not adjacent to the two empty seats. From my earlier reasoning, I think I accounted for that by excluding the adjacent seats when choosing the third empty seat.But let me think about another approach to confirm. Maybe using the inclusion-exclusion principle.First, calculate the total number of ways to seat 3 people in 6 seats without any restrictions. That would be C(6,3) * 3! = 20 * 6 = 120.Now, from this total, subtract the arrangements where all three empty seats are adjacent and the arrangements where all three empty seats are separated.First, arrangements with all three empty seats adjacent. Treating the three empty seats as a single unit, we have 4 units to arrange: the block of three empty seats and the three people. The number of ways to arrange these is C(4,1) * 3! = 4 * 6 = 24. Wait, but actually, the number of ways to place the block of three empty seats is 4: positions (1,2,3), (2,3,4), (3,4,5), (4,5,6). For each of these, the people can be arranged in the remaining 3 seats in 3! = 6 ways. So, total arrangements with all three empty seats adjacent: 4 * 6 = 24.Next, arrangements where all three empty seats are separated. This means no two empty seats are adjacent. To calculate this, we can use the concept of placing the three people first and then inserting the empty seats in the gaps.If we have 3 people seated, there are 4 gaps where the empty seats can be placed: before the first person, between the first and second, between the second and third, and after the third person. We need to choose 3 gaps out of these 4 to place one empty seat each. The number of ways to do this is C(4,3) = 4. For each of these, the people can be arranged in 3! = 6 ways. So, total arrangements with all three empty seats separated: 4 * 6 = 24.Therefore, the number of arrangements where exactly two empty seats are adjacent is total arrangements minus arrangements with all three empty seats adjacent minus arrangements with all three empty seats separated: 120 - 24 - 24 = 72.Hmm, that matches my earlier result. So, the answer should be 72, which is option C.But wait, let me think again. Is there a possibility that when I subtracted the arrangements with all three empty seats adjacent and separated, I might have missed something? For example, are there any overlapping cases or something else?No, because the total arrangements are 120, and the only possibilities for the empty seats are: all three adjacent, exactly two adjacent, or all three separated. There's no overlap between these cases, so subtracting them should give the correct number of arrangements with exactly two adjacent empty seats.Therefore, I'm confident that the answer is 72, which is option C.

answer：Okay, so I have this problem about geometric sequences. Let me try to understand it step by step. The problem says:"Given that the sequence {a_n} is a geometric sequence with a₁ = 4 and common ratio q. The sum of the first n terms is S_n. If the sequence {S_n + 2} is also a geometric sequence, then q = (　). The options are A: 2, B: -2, C: 3, D: -3."Alright, so first, let me recall what a geometric sequence is. A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio, which is q in this case.Given that a₁ = 4, so the first term is 4. The common ratio is q, so the sequence goes 4, 4q, 4q², 4q³, and so on.Now, the sum of the first n terms of a geometric sequence is given by the formula:S_n = a₁(1 - qⁿ)/(1 - q) if q ≠ 1.Since q is a common ratio, it's possible that q could be 1, but in that case, the sum would just be S_n = a₁ * n. However, since the problem doesn't specify q ≠ 1, I should keep that in mind, but I think in this case, q is not 1 because otherwise, the sum would be linear, and adding 2 to it might not make it a geometric sequence.But let's proceed step by step.The problem states that {S_n + 2} is also a geometric sequence. So, if I take each term S_n, add 2 to it, and then the resulting sequence is geometric. That means that the ratio between consecutive terms of {S_n + 2} should be constant.So, for {S_n + 2} to be a geometric sequence, the ratio (S_{n+1} + 2)/(S_n + 2) should be constant for all n.Let me write that down:(S_{n+1} + 2)/(S_n + 2) = constant ratio, say r.So, for all n, this ratio should be equal to r.Alternatively, another way to think about it is that the sequence {S_n + 2} satisfies the property of a geometric sequence, so the square of the middle term is equal to the product of the terms on either side. That is, for any three consecutive terms, (S_{n+1} + 2)² = (S_n + 2)(S_{n+2} + 2).This is similar to the property that in a geometric sequence, the square of the middle term is equal to the product of the adjacent terms.So, maybe I can use this property to find q.Let me try to compute S₁, S₂, S₃, etc., and then see what conditions arise from {S_n + 2} being a geometric sequence.First, let's compute S₁, S₂, S₃.Given that a₁ = 4, a₂ = 4q, a₃ = 4q², and so on.So, S₁ = a₁ = 4.S₂ = a₁ + a₂ = 4 + 4q.S₃ = a₁ + a₂ + a₃ = 4 + 4q + 4q².Similarly, S₄ = 4 + 4q + 4q² + 4q³, and so on.Now, the sequence {S_n + 2} is supposed to be geometric. So, let's compute the first few terms of {S_n + 2}:(S₁ + 2) = 4 + 2 = 6(S₂ + 2) = (4 + 4q) + 2 = 6 + 4q(S₃ + 2) = (4 + 4q + 4q²) + 2 = 6 + 4q + 4q²(S₄ + 2) = (4 + 4q + 4q² + 4q³) + 2 = 6 + 4q + 4q² + 4q³And so on.Since {S_n + 2} is a geometric sequence, the ratio between consecutive terms should be constant. So, the ratio between (S₂ + 2) and (S₁ + 2) should be equal to the ratio between (S₃ + 2) and (S₂ + 2), and so on.So, let's compute the ratio r:r = (S₂ + 2)/(S₁ + 2) = (6 + 4q)/6 = 1 + (4q)/6 = 1 + (2q)/3Similarly, the ratio between (S₃ + 2) and (S₂ + 2) should also be r:r = (S₃ + 2)/(S₂ + 2) = (6 + 4q + 4q²)/(6 + 4q)So, setting these equal:(6 + 4q + 4q²)/(6 + 4q) = 1 + (2q)/3Let me write that equation:(6 + 4q + 4q²)/(6 + 4q) = 1 + (2q)/3Let me simplify the left-hand side (LHS):LHS = (6 + 4q + 4q²)/(6 + 4q) = [4q² + 4q + 6]/[4q + 6]I can factor numerator and denominator:Numerator: 4q² + 4q + 6 = 2(2q² + 2q + 3)Denominator: 4q + 6 = 2(2q + 3)So, LHS = [2(2q² + 2q + 3)]/[2(2q + 3)] = (2q² + 2q + 3)/(2q + 3)So, LHS simplifies to (2q² + 2q + 3)/(2q + 3)Now, the right-hand side (RHS) is 1 + (2q)/3 = (3 + 2q)/3So, setting LHS = RHS:(2q² + 2q + 3)/(2q + 3) = (3 + 2q)/3Cross-multiplying:3(2q² + 2q + 3) = (2q + 3)(3 + 2q)Let me compute both sides.Left side: 3*(2q² + 2q + 3) = 6q² + 6q + 9Right side: (2q + 3)(3 + 2q) = (2q + 3)^2 = 4q² + 12q + 9So, equation becomes:6q² + 6q + 9 = 4q² + 12q + 9Subtracting 4q² + 12q + 9 from both sides:6q² + 6q + 9 - 4q² - 12q - 9 = 0Simplify:(6q² - 4q²) + (6q - 12q) + (9 - 9) = 02q² - 6q + 0 = 0So, 2q² - 6q = 0Factor out 2q:2q(q - 3) = 0So, 2q = 0 or q - 3 = 0Thus, q = 0 or q = 3But q = 0 would make the geometric sequence {a_n} collapse after the first term, which is not typical because in a geometric sequence, the common ratio is usually non-zero. Also, if q = 0, then S_n would be 4 for all n ≥ 1, so {S_n + 2} would be 6, 6, 6, ..., which is a constant sequence, technically a geometric sequence with common ratio 1. However, in the problem, the options do not include q = 0, so we can disregard that.Therefore, q = 3.Wait, but let me verify this because sometimes when we square terms or cross-multiply, we might introduce extraneous solutions.Let me check if q = 3 satisfies the original condition.Compute S₁, S₂, S₃:S₁ = 4S₂ = 4 + 4*3 = 4 + 12 = 16S₃ = 4 + 12 + 36 = 52Now, {S_n + 2} would be:6, 18, 54, ...Which is 6, 18, 54,...Check if this is a geometric sequence.Compute the ratios:18/6 = 354/18 = 3So, yes, the ratio is 3, which is consistent. So, {S_n + 2} is indeed a geometric sequence with common ratio 3 when q = 3.Therefore, q = 3 is the correct answer, which is option C.But wait, let me check if there's another possible value for q. Earlier, we had q = 0 or q = 3. But q = 0 is not in the options, so q = 3 is the only solution.Alternatively, let me think if there's another approach to solve this problem.Another way is to consider that {S_n + 2} is a geometric sequence, so the difference between consecutive terms should also form a geometric sequence, but that might complicate things.Alternatively, since {S_n + 2} is geometric, the ratio between terms should be constant.So, for all n, (S_{n+1} + 2)/(S_n + 2) = r, where r is the common ratio.But S_{n+1} = S_n + a_{n+1}, so:(S_n + a_{n+1} + 2)/(S_n + 2) = rWhich simplifies to:1 + (a_{n+1})/(S_n + 2) = rSo, (a_{n+1})/(S_n + 2) = r - 1But a_{n+1} = a₁*q^n = 4*q^nSo, 4*q^n / (S_n + 2) = r - 1But S_n is the sum of the first n terms of the geometric sequence:S_n = 4*(1 - q^n)/(1 - q)So, S_n + 2 = 4*(1 - q^n)/(1 - q) + 2Let me write that:S_n + 2 = [4*(1 - q^n) + 2*(1 - q)] / (1 - q)Simplify numerator:4 - 4q^n + 2 - 2q = (4 + 2) - 4q^n - 2q = 6 - 4q^n - 2qSo, S_n + 2 = (6 - 4q^n - 2q)/(1 - q)So, going back to the equation:4*q^n / (S_n + 2) = r - 1Substitute S_n + 2:4*q^n / [(6 - 4q^n - 2q)/(1 - q)] = r - 1Which is equal to:4*q^n*(1 - q)/(6 - 4q^n - 2q) = r - 1Hmm, this seems a bit complicated, but maybe I can find a relationship here.Let me denote r - 1 as k, so:4*q^n*(1 - q)/(6 - 4q^n - 2q) = kBut this equation must hold for all n, which suggests that the left-hand side must be independent of n. That is, the dependence on q^n must cancel out.Looking at the numerator: 4*q^n*(1 - q)Denominator: 6 - 4q^n - 2qSo, to have the ratio independent of n, the coefficients of q^n in numerator and denominator must be proportional.Let me write the denominator as: (-4q^n) + (6 - 2q)Similarly, numerator is: 4*(1 - q)*q^nSo, for the ratio to be independent of n, the coefficients of q^n must be proportional, and the constants must also be proportional.So, let's set up the proportion:Numerator: 4*(1 - q)*q^nDenominator: (-4)q^n + (6 - 2q)So, for the ratio to be independent of n, the ratio of the coefficients of q^n must equal the ratio of the constants.That is:[4*(1 - q)] / (-4) = (6 - 2q)/ (something)Wait, actually, let me think differently.Let me suppose that the ratio is k, so:4*(1 - q)*q^n / (-4q^n + (6 - 2q)) = kMultiply both sides by denominator:4*(1 - q)*q^n = k*(-4q^n + 6 - 2q)Let me rearrange:4*(1 - q)*q^n = -4k q^n + k*(6 - 2q)Now, collect like terms:[4*(1 - q) + 4k] q^n = k*(6 - 2q)Since this equation must hold for all n, the coefficients of q^n must be zero unless the right-hand side is also a multiple of q^n, but the right-hand side is a constant. Therefore, the coefficient of q^n on the left must be zero, and the constant term must equal the right-hand side.So, set coefficients equal:Coefficient of q^n: 4*(1 - q) + 4k = 0Constant term: 0 = k*(6 - 2q)Wait, but the left-hand side has no constant term, so the right-hand side must also be zero. Therefore:k*(6 - 2q) = 0So, either k = 0 or 6 - 2q = 0.Case 1: k = 0If k = 0, then from the coefficient equation:4*(1 - q) + 4*0 = 0 => 4*(1 - q) = 0 => 1 - q = 0 => q = 1But if q = 1, then the original geometric sequence {a_n} would have all terms equal to 4, so S_n = 4n.Then, {S_n + 2} would be 4n + 2, which is a linear sequence, not a geometric sequence unless the common ratio is 1, but 4n + 2 is not a geometric sequence because the ratio between terms is not constant. For example, (4*2 + 2)/(4*1 + 2) = 10/6 = 5/3, and (4*3 + 2)/(4*2 + 2) = 14/10 = 7/5, which are not equal. So, q = 1 is not a valid solution.Case 2: 6 - 2q = 0 => 6 = 2q => q = 3So, q = 3.Now, let's check the coefficient equation with q = 3:4*(1 - 3) + 4k = 0 => 4*(-2) + 4k = 0 => -8 + 4k = 0 => 4k = 8 => k = 2So, k = 2, which means r - 1 = 2 => r = 3So, the common ratio of {S_n + 2} is 3, which we already verified earlier.Therefore, q = 3 is the solution.So, the answer is C: 3.**Final Answer**boxed{C}