answer：Alright, so I have this problem where I need to find the complex number α, given that both α + β and i(α - 3β) are positive real numbers. They also told me that β is 2 + 3i. Hmm, okay, let me try to unpack this step by step.First, let me recall that a complex number has a real part and an imaginary part. So, α can be written as a + bi, where a and b are real numbers. Similarly, β is given as 2 + 3i, so its real part is 2 and the imaginary part is 3.Now, the problem says that α + β is a positive real number. Let me write that out:α + β = (a + bi) + (2 + 3i) = (a + 2) + (b + 3)i.Since this sum is a positive real number, the imaginary part must be zero. That means:b + 3 = 0 ⇒ b = -3.So, the imaginary part of α is -3. That's one piece of information.Next, the problem also states that i(α - 3β) is a positive real number. Let me compute α - 3β first.α - 3β = (a + bi) - 3*(2 + 3i) = (a - 6) + (b - 9)i.Now, multiplying this by i:i*(α - 3β) = i*((a - 6) + (b - 9)i) = i*(a - 6) + i*(b - 9)i.I know that i*i = -1, so this becomes:= (a - 6)i + (b - 9)*(-1) = -(b - 9) + (a - 6)i.Simplify that:= (9 - b) + (a - 6)i.Now, since i(α - 3β) is a positive real number, the imaginary part must be zero. So:a - 6 = 0 ⇒ a = 6.So, the real part of α is 6.Putting it all together, α has a real part of 6 and an imaginary part of -3. Therefore, α is:α = 6 - 3i.Wait, let me double-check to make sure I didn't make any mistakes.First, check α + β:α + β = (6 - 3i) + (2 + 3i) = 8 + 0i, which is indeed a positive real number. Good.Next, check i(α - 3β):First, compute α - 3β:α - 3β = (6 - 3i) - 3*(2 + 3i) = (6 - 3i) - (6 + 9i) = 6 - 3i - 6 - 9i = -12i.Then, multiply by i:i*(-12i) = -12i² = -12*(-1) = 12, which is a positive real number. Perfect.So, both conditions are satisfied. Therefore, α is indeed 6 - 3i.I think that's it. I don't see any mistakes in my calculations, and the checks confirm that the solution works. So, I'm confident that α is 6 - 3i.**Final Answer**The value of α is boxed{6 - 3i}.

answer：Okay, so I have this problem where I need to figure out the maximum number of points, n, on a plane such that any three points form a right triangle. Hmm, interesting. Let me try to break this down step by step.First, I know that a right triangle has one 90-degree angle. So, for any three points I pick, one of the angles has to be right. That means the points have to be arranged in a way that no matter which three I choose, they always satisfy the condition of forming a right triangle.Let me start with smaller numbers to get a sense of what's going on. If n is 3, obviously, they form a triangle, and if it's a right triangle, that's fine. But the question is about the maximum n where this condition holds for any three points.What about n=4? Let me think. If I place four points at the corners of a square, then any three points will form a right triangle. For example, picking any three corners of a square will give me a right triangle because the square's sides are perpendicular. So, n=4 seems to work.But can we go higher? Let's try n=5. If I add another point, where can I place it so that any three points still form a right triangle? Hmm, this is tricky. Let me visualize. If I have a square and add a fifth point somewhere, say inside the square or outside, will that maintain the condition?Suppose I place the fifth point at the center of the square. Now, if I pick three points: two corners and the center, does that form a right triangle? Let's see. The center point is equidistant from all corners, but the angles might not necessarily be 90 degrees. For example, if I take two adjacent corners and the center, the triangle formed might not be right-angled because the distances from the center to the corners are equal, but the angles could be different.Alternatively, if I place the fifth point somewhere else, maybe on the extension of one of the square's sides. But then, when I pick three points that include this new point and two others, will they form a right triangle? It's not clear. There's a risk that some triangles might not have a right angle.Wait, maybe there's a different configuration for five points where every three points form a right triangle. Let me think about circles. If I have points on a circle, any triangle inscribed in a semicircle is a right triangle. So, if all points lie on a semicircle, any three points will form a right triangle. But can I have five points on a semicircle?Yes, but then all five points would lie on a semicircle, and any three points would form a right triangle. But wait, if all five points are on a semicircle, then actually, any three points would form a right triangle because of Thales' theorem. So, does that mean n=5 is possible?Hold on, no. Because if all five points are on a semicircle, then any three points will indeed form a right triangle, but the problem is that the points are on a plane, not necessarily on a semicircle. So, if I arrange five points on a semicircle, that's a valid configuration, but is that the only way?Wait, but the problem says "on a plane," so arranging points on a semicircle is allowed. So, maybe n=5 is possible? But I'm not sure because when I think about it, if all five points are on a semicircle, then any three points will form a right triangle. But actually, no, because if you take three points on a semicircle, the triangle is right-angled only if one of the sides is the diameter. So, if all five points are on a semicircle, then any three points will have one side as the diameter, hence forming a right triangle.Wait, but in reality, if you have five points on a semicircle, not all triples will necessarily have the diameter as one of their sides. For example, if you take three points that are not including the endpoints of the diameter, then the triangle might not be right-angled. Hmm, so maybe my initial thought was wrong.Let me clarify. Thales' theorem states that if you have a triangle inscribed in a semicircle, then it's a right triangle. But for that, one side has to be the diameter. So, if I have five points on a semicircle, not all triples will include the diameter. Therefore, not all triples will form a right triangle. So, n=5 might not be possible.Wait, but if I arrange the five points such that every pair of points is part of a diameter? That seems impossible because each point can only be paired with one other point as a diameter. So, with five points, you can have at most two diameters, leaving one point without a pair. Therefore, not all triples will include a diameter, so not all triples will be right triangles.Therefore, n=5 is not possible because we can't arrange five points on a plane such that every three points form a right triangle.Wait, but earlier I thought that four points on a square work because any three points form a right triangle. So, maybe four is the maximum.Let me confirm. For four points on a square, any three points will form a right triangle. For example, picking three corners of the square, you get a right triangle. If you pick two adjacent corners and the opposite corner, you still get a right triangle. So, yes, four points work.But when I try to add a fifth point, I can't find a configuration where every three points form a right triangle. Whether I place it inside, outside, or on a semicircle, there will always be some triples that don't form right triangles.Therefore, the maximum n is 4.Wait, but let me think again. Maybe there's another configuration besides the square. What if I arrange the points in a different way? For example, three points forming a right triangle, and the fourth point somewhere else. But then, when I pick three points that include the fourth point, will they form a right triangle?Suppose I have three points forming a right triangle, and the fourth point is the orthocenter or something. But I'm not sure. It might not necessarily form right triangles with all combinations.Alternatively, maybe arranging four points as the vertices of a rectangle. Wait, a rectangle is similar to a square in that any three points form a right triangle. So, that's another configuration.But again, adding a fifth point seems problematic.So, after considering different configurations and realizing that adding a fifth point breaks the condition, I think the maximum n is 4.

answer：First, convert all amounts to cents for easier calculation.Lance has 70 cents.Margaret has three-fourths of a dollar, which is 75 cents.Guy has two quarters and a dime, totaling 60 cents.Add up the amounts of Lance, Margaret, and Guy:70 cents (Lance) + 75 cents (Margaret) + 60 cents (Guy) = 205 cents.Subtract this total from the combined total of 265 cents to find out how much Bill has:265 cents - 205 cents = 60 cents.Therefore, Bill has 60 cents. Without specific information about the denominations, there are multiple possibilities for the coins Bill could have.

answer：Okay, so I have this problem about a hyperbola. Let me try to understand it step by step. The problem says that the length of the real axis of a hyperbola is m. Hmm, I remember that in a hyperbola, the real axis is also called the transverse axis. The length of the transverse axis is 2a, where a is the distance from the center to each vertex. So, if the length is m, that means 2a = m, right? So, a = m/2.Now, the problem also mentions a point P on the hyperbola. The distance from P to the right focus is m. I need to find the distance from P to the left focus. I think I remember that for hyperbolas, the difference of distances from any point on the hyperbola to the two foci is constant and equal to 2a. So, if I denote the distance from P to the right focus as d1 and the distance to the left focus as d2, then |d1 - d2| = 2a.Given that the distance from P to the right focus is m, so d1 = m. Since 2a = m, then the difference |d1 - d2| should be equal to m. So, |m - d2| = m. Hmm, let me think about this. If I take the absolute value, it can be either m - d2 = m or d2 - m = m. Let's consider both cases. First case: m - d2 = m. If I subtract m from both sides, I get -d2 = 0, which implies d2 = 0. But that doesn't make sense because the distance from a point on the hyperbola to a focus can't be zero unless the point is the focus itself, which isn't the case here.Second case: d2 - m = m. If I add m to both sides, I get d2 = 2m. That seems reasonable. So, the distance from P to the left focus is 2m.Wait, let me double-check. The definition of a hyperbola is the set of all points where the absolute difference of the distances to the two foci is constant, which is 2a. In this case, 2a is m, so the difference should be m. Since d1 is m, then d2 must be m + m = 2m. Yeah, that makes sense.I think I got it. So, the distance from point P to the left focus is 2m.

answer：First, determine how many miles were paved on the second day. The company paved 4 miles on the first day, so one mile less than double that would be:[(4 times 2) - 1 = 8 - 1 = 7 text{ miles}]Next, calculate the total miles paved after two days:[4 text{ miles} + 7 text{ miles} = 11 text{ miles}]The total length of the road is 16 miles, so the remaining road to be paved on the third day is:[16 text{ miles} - 11 text{ miles} = 5 text{ miles}]Since it takes three truckloads of asphalt to pave each mile of road, for 5 miles, the company will need:[5 text{ miles} times 3 text{ truckloads/mile} = 15 text{ truckloads}]Each truckload of asphalt uses five times as many bags of gravel as barrels of pitch. Given that each truckload uses two bags of gravel, the number of barrels of pitch per truckload is:[frac{2 text{ bags of gravel}}{5} = 0.4 text{ barrels of pitch}]Therefore, for 15 truckloads, the company will need:[15 text{ truckloads} times 0.4 text{ barrels/truckload} = 6 text{ barrels of pitch}]

answer：Okay, so I have this problem about a parabola, a chord passing through the focus, and some surface areas created by rotating that chord around the directrix. Hmm, let me try to break this down step by step.First, the equation of the parabola is given as ( y^2 = 2px ). I remember that for a parabola in the form ( y^2 = 4ax ), the focus is at ( (a, 0) ) and the directrix is the line ( x = -a ). Comparing this to our equation ( y^2 = 2px ), it seems like ( 4a = 2p ), so ( a = frac{p}{2} ). That means the focus is at ( left( frac{p}{2}, 0 right) ) and the directrix is the line ( x = -frac{p}{2} ).Now, the problem says that ( PQ ) is any chord passing through the focus. So, ( PQ ) is a line segment that goes through the focus ( left( frac{p}{2}, 0 right) ). I need to find the projection of this chord ( PQ ) onto the directrix ( l ), which is the line ( x = -frac{p}{2} ). The projection is called ( MN ).Next, I have to find two surface areas: ( S_1 ) is the surface area obtained by rotating ( PQ ) around the directrix ( l ), and ( S_2 ) is the surface area of the sphere with ( MN ) as its diameter. Then, I need to compare ( S_1 ) and ( S_2 ) and choose the correct conclusion from the options given.Let me visualize this. Rotating ( PQ ) around the directrix ( l ) will create a kind of surface, probably a cylinder or something similar. On the other hand, a sphere with ( MN ) as its diameter will have a surface area based on the length of ( MN ).I think I need to find expressions for both ( S_1 ) and ( S_2 ) in terms of the length of ( PQ ) or some other parameter, and then compare them.First, let me find the length of ( PQ ). Since ( PQ ) is a chord passing through the focus, I can use the property of parabolas that the length of a focal chord is related to the parameters of the parabola. For a parabola ( y^2 = 4ax ), the length of a focal chord making an angle ( theta ) with the x-axis is ( frac{4a}{sin^2 theta} ). But in our case, the parabola is ( y^2 = 2px ), so ( 4a = 2p ) which means ( a = frac{p}{2} ). Therefore, the length of the focal chord would be ( frac{4a}{sin^2 theta} = frac{2p}{sin^2 theta} ). Hmm, but I don't know the angle ( theta ) here. Maybe there's another way.Alternatively, maybe I can parametrize the points ( P ) and ( Q ) on the parabola. Let's say ( P ) is ( (x_1, y_1) ) and ( Q ) is ( (x_2, y_2) ). Since both points lie on the parabola, they satisfy ( y_1^2 = 2p x_1 ) and ( y_2^2 = 2p x_2 ). Also, since ( PQ ) passes through the focus ( left( frac{p}{2}, 0 right) ), the equation of the chord ( PQ ) must satisfy this point.The equation of the chord can be written in the two-point form. The slope ( m ) of ( PQ ) is ( frac{y_2 - y_1}{x_2 - x_1} ). Since it passes through ( left( frac{p}{2}, 0 right) ), the equation is ( y - y_1 = m left( x - x_1 right) ). Plugging in ( x = frac{p}{2} ) and ( y = 0 ), we get:( 0 - y_1 = m left( frac{p}{2} - x_1 right) )Substituting ( m ):( -y_1 = frac{y_2 - y_1}{x_2 - x_1} left( frac{p}{2} - x_1 right) )This seems a bit complicated. Maybe there's a property of parabolas that I can use here. I recall that for any focal chord, the product of the distances from the focus to each end is equal to the square of the focal length. Wait, not sure about that.Alternatively, maybe I can use parametric coordinates for the parabola. For ( y^2 = 4ax ), the parametric equations are ( x = at^2 ), ( y = 2at ). For our parabola ( y^2 = 2px ), comparing, we have ( 4a = 2p ) so ( a = frac{p}{2} ). Therefore, the parametric equations would be ( x = frac{p}{2} t^2 ), ( y = p t ).So, let me denote point ( P ) as ( left( frac{p}{2} t^2, p t right) ) and point ( Q ) as ( left( frac{p}{2} s^2, p s right) ). Since ( PQ ) is a focal chord, it passes through the focus ( left( frac{p}{2}, 0 right) ). So, the line joining ( P ) and ( Q ) must pass through ( left( frac{p}{2}, 0 right) ).The equation of the line PQ can be found using the two points ( P ) and ( Q ). The slope ( m ) is:( m = frac{p s - p t}{frac{p}{2} s^2 - frac{p}{2} t^2} = frac{p (s - t)}{frac{p}{2} (s^2 - t^2)} = frac{2 (s - t)}{(s - t)(s + t)} = frac{2}{s + t} )So, the slope is ( frac{2}{s + t} ). Now, the equation of the line is:( y - p t = frac{2}{s + t} left( x - frac{p}{2} t^2 right) )Since this line passes through ( left( frac{p}{2}, 0 right) ), substituting ( x = frac{p}{2} ) and ( y = 0 ):( 0 - p t = frac{2}{s + t} left( frac{p}{2} - frac{p}{2} t^2 right) )Simplify the right side:( frac{2}{s + t} cdot frac{p}{2} (1 - t^2) = frac{p (1 - t^2)}{s + t} )So, we have:( -p t = frac{p (1 - t^2)}{s + t} )Divide both sides by ( p ):( -t = frac{1 - t^2}{s + t} )Multiply both sides by ( s + t ):( -t (s + t) = 1 - t^2 )Expand the left side:( -s t - t^2 = 1 - t^2 )Add ( t^2 ) to both sides:( -s t = 1 )So, ( s = -frac{1}{t} )Therefore, the parameter ( s ) is related to ( t ) by ( s = -frac{1}{t} ). So, if I choose a parameter ( t ), then ( s ) is determined as ( -1/t ).Now, let's find the coordinates of ( P ) and ( Q ):Point ( P ): ( left( frac{p}{2} t^2, p t right) )Point ( Q ): ( left( frac{p}{2} s^2, p s right) = left( frac{p}{2} left( frac{1}{t^2} right), p left( -frac{1}{t} right) right) = left( frac{p}{2 t^2}, -frac{p}{t} right) )So, ( P ) is ( left( frac{p}{2} t^2, p t right) ) and ( Q ) is ( left( frac{p}{2 t^2}, -frac{p}{t} right) ).Now, let's compute the length of chord ( PQ ). The distance between ( P ) and ( Q ) is:( |PQ| = sqrt{ left( frac{p}{2 t^2} - frac{p}{2} t^2 right)^2 + left( -frac{p}{t} - p t right)^2 } )Simplify the x-coordinate difference:( frac{p}{2 t^2} - frac{p}{2} t^2 = frac{p}{2} left( frac{1}{t^2} - t^2 right) = frac{p}{2} cdot frac{1 - t^4}{t^2} )The y-coordinate difference:( -frac{p}{t} - p t = -p left( frac{1}{t} + t right) = -p cdot frac{1 + t^2}{t} )So, the length squared is:( |PQ|^2 = left( frac{p}{2} cdot frac{1 - t^4}{t^2} right)^2 + left( -p cdot frac{1 + t^2}{t} right)^2 )Factor out ( p^2 ):( |PQ|^2 = p^2 left[ left( frac{1 - t^4}{2 t^2} right)^2 + left( frac{1 + t^2}{t} right)^2 right] )Let me compute each term:First term:( left( frac{1 - t^4}{2 t^2} right)^2 = frac{(1 - t^4)^2}{4 t^4} )Second term:( left( frac{1 + t^2}{t} right)^2 = frac{(1 + t^2)^2}{t^2} )So, putting it together:( |PQ|^2 = p^2 left[ frac{(1 - t^4)^2}{4 t^4} + frac{(1 + t^2)^2}{t^2} right] )This looks complicated. Maybe I can simplify it.First, note that ( 1 - t^4 = (1 - t^2)(1 + t^2) ). So,( (1 - t^4)^2 = (1 - t^2)^2 (1 + t^2)^2 )Therefore, the first term becomes:( frac{(1 - t^2)^2 (1 + t^2)^2}{4 t^4} )The second term is:( frac{(1 + t^2)^2}{t^2} )Let me factor out ( frac{(1 + t^2)^2}{t^4} ) from both terms:( |PQ|^2 = p^2 cdot frac{(1 + t^2)^2}{t^4} left[ frac{(1 - t^2)^2}{4} + t^2 right] )Compute the expression inside the brackets:( frac{(1 - t^2)^2}{4} + t^2 = frac{1 - 2 t^2 + t^4}{4} + t^2 = frac{1 - 2 t^2 + t^4 + 4 t^2}{4} = frac{1 + 2 t^2 + t^4}{4} = frac{(1 + t^2)^2}{4} )So, putting it back:( |PQ|^2 = p^2 cdot frac{(1 + t^2)^2}{t^4} cdot frac{(1 + t^2)^2}{4} = p^2 cdot frac{(1 + t^2)^4}{4 t^4} )Therefore, ( |PQ| = p cdot frac{(1 + t^2)^2}{2 t^2} )Simplify:( |PQ| = frac{p (1 + t^2)^2}{2 t^2} )Okay, so that's the length of chord ( PQ ).Now, moving on to the projection ( MN ) of ( PQ ) onto the directrix ( l ), which is the line ( x = -frac{p}{2} ).Projection of a line segment onto another line can be found by considering the component of the segment in the direction perpendicular to the line. Since the directrix is a vertical line, the projection of ( PQ ) onto it will be the horizontal component of ( PQ ).Wait, no. Actually, the projection of a segment onto a line is the length of the shadow of the segment when light is shone perpendicular to the line. Since the directrix is vertical, the projection of ( PQ ) onto it would involve the horizontal component.But actually, since the directrix is a vertical line, the projection of ( PQ ) onto it would be the set of points on the directrix closest to ( PQ ). So, the projection ( MN ) is the image of ( PQ ) on the directrix.Alternatively, since the directrix is vertical, the projection of ( PQ ) onto it would be the segment between the projections of ( P ) and ( Q ) onto the directrix.The projection of a point ( (x, y) ) onto the directrix ( x = -frac{p}{2} ) is the point ( left( -frac{p}{2}, y right) ). So, the projection of ( P ) is ( left( -frac{p}{2}, p t right) ) and the projection of ( Q ) is ( left( -frac{p}{2}, -frac{p}{t} right) ).Therefore, the projection ( MN ) is the segment between ( left( -frac{p}{2}, p t right) ) and ( left( -frac{p}{2}, -frac{p}{t} right) ). So, the length of ( MN ) is the distance between these two points, which is:( |MN| = sqrt{ left( -frac{p}{2} - left( -frac{p}{2} right) right)^2 + left( -frac{p}{t} - p t right)^2 } = sqrt{0 + left( -frac{p}{t} - p t right)^2 } = | -frac{p}{t} - p t | = p left| frac{1}{t} + t right| )Since ( t ) is a real number (except 0), and the parabola is symmetric, we can assume ( t > 0 ) without loss of generality. So,( |MN| = p left( frac{1}{t} + t right) )Okay, so ( |MN| = p left( t + frac{1}{t} right) )Now, moving on to ( S_1 ), which is the surface area obtained by rotating ( PQ ) around the directrix ( l ).Since the directrix is the line ( x = -frac{p}{2} ), and ( PQ ) is a chord in the plane, rotating ( PQ ) around the directrix will create a surface of revolution. The surface area can be calculated using the formula for the surface of revolution:( S = 2 pi int_{a}^{b} r(x) sqrt{1 + [f'(x)]^2} dx )But since ( PQ ) is a straight line segment, maybe it's easier to think of it as a line segment being rotated around an axis, which would form a kind of truncated cone or a cylinder, depending on the orientation.Wait, actually, since ( PQ ) is a straight line, rotating it around the directrix ( l ) will create a surface that is a portion of a cone or a cylinder. But since ( l ) is vertical, and ( PQ ) is a chord, the surface will be a kind of conical surface.Alternatively, maybe it's a cylinder if the distance from ( PQ ) to ( l ) is constant. Wait, no, because ( PQ ) is a chord, so the distance from each point on ( PQ ) to ( l ) varies.Wait, perhaps I should parametrize the line segment ( PQ ) and compute the surface area by integrating.Let me parametrize ( PQ ) from point ( P ) to point ( Q ). Let me use a parameter ( lambda ) that goes from 0 to 1, where ( lambda = 0 ) corresponds to ( P ) and ( lambda = 1 ) corresponds to ( Q ).So, the parametric equations for ( PQ ) are:( x(lambda) = x_P + lambda (x_Q - x_P) )( y(lambda) = y_P + lambda (y_Q - y_P) )From earlier, we have:( x_P = frac{p}{2} t^2 ), ( y_P = p t )( x_Q = frac{p}{2 t^2} ), ( y_Q = -frac{p}{t} )So,( x(lambda) = frac{p}{2} t^2 + lambda left( frac{p}{2 t^2} - frac{p}{2} t^2 right) = frac{p}{2} t^2 + lambda cdot frac{p}{2} left( frac{1}{t^2} - t^2 right) )( y(lambda) = p t + lambda left( -frac{p}{t} - p t right) = p t - lambda p left( frac{1}{t} + t right) )Now, the distance from a point ( (x, y) ) on ( PQ ) to the directrix ( l ) (which is ( x = -frac{p}{2} )) is ( |x - (-frac{p}{2})| = |x + frac{p}{2}| ). Since ( x ) is always greater than or equal to ( -frac{p}{2} ) for points on the parabola, this distance is ( x + frac{p}{2} ).Therefore, the radius of revolution at each point ( lambda ) is ( r(lambda) = x(lambda) + frac{p}{2} ).So, ( r(lambda) = frac{p}{2} t^2 + frac{p}{2} left( frac{1}{t^2} - t^2 right) lambda + frac{p}{2} )Simplify:( r(lambda) = frac{p}{2} t^2 + frac{p}{2} cdot frac{1 - t^4}{t^2} lambda + frac{p}{2} )Wait, maybe I should compute ( x(lambda) + frac{p}{2} ):( x(lambda) + frac{p}{2} = frac{p}{2} t^2 + lambda cdot frac{p}{2} left( frac{1}{t^2} - t^2 right) + frac{p}{2} )Factor out ( frac{p}{2} ):( x(lambda) + frac{p}{2} = frac{p}{2} left[ t^2 + lambda left( frac{1}{t^2} - t^2 right) + 1 right] )Let me compute the derivative ( frac{dr}{dlambda} ):Wait, actually, for the surface area, we need the differential arc length ( ds ) along ( PQ ), which is given by:( ds = sqrt{ left( frac{dx}{dlambda} right)^2 + left( frac{dy}{dlambda} right)^2 } dlambda )Compute ( frac{dx}{dlambda} ):( frac{dx}{dlambda} = frac{p}{2} left( frac{1}{t^2} - t^2 right) )Compute ( frac{dy}{dlambda} ):( frac{dy}{dlambda} = -p left( frac{1}{t} + t right) )So,( ds = sqrt{ left( frac{p}{2} left( frac{1}{t^2} - t^2 right) right)^2 + left( -p left( frac{1}{t} + t right) right)^2 } dlambda )Factor out ( p^2 ):( ds = p sqrt{ left( frac{1}{2} left( frac{1}{t^2} - t^2 right) right)^2 + left( frac{1}{t} + t right)^2 } dlambda )Let me compute the expression inside the square root:First term:( left( frac{1}{2} left( frac{1}{t^2} - t^2 right) right)^2 = frac{1}{4} left( frac{1}{t^4} - 2 + t^4 right) )Second term:( left( frac{1}{t} + t right)^2 = frac{1}{t^2} + 2 + t^2 )So, adding them together:( frac{1}{4} left( frac{1}{t^4} - 2 + t^4 right) + frac{1}{t^2} + 2 + t^2 )Simplify:( frac{1}{4 t^4} - frac{1}{2} + frac{t^4}{4} + frac{1}{t^2} + 2 + t^2 )Combine like terms:- Constants: ( -frac{1}{2} + 2 = frac{3}{2} )- ( frac{1}{t^4} ) term: ( frac{1}{4 t^4} )- ( frac{1}{t^2} ) term: ( frac{1}{t^2} )- ( t^4 ) term: ( frac{t^4}{4} )- ( t^2 ) term: ( t^2 )So, the expression becomes:( frac{1}{4 t^4} + frac{1}{t^2} + t^2 + frac{t^4}{4} + frac{3}{2} )Hmm, this seems complicated. Maybe there's a better way.Wait, earlier I found that ( |PQ| = frac{p (1 + t^2)^2}{2 t^2} ). So, the length of ( PQ ) is ( frac{p (1 + t^2)^2}{2 t^2} ). Therefore, the differential arc length ( ds ) is just ( |PQ| dlambda ), since ( lambda ) goes from 0 to 1.Wait, no. Actually, ( ds = |PQ| dlambda ) because ( lambda ) is a parameter that goes from 0 to 1 as we move from ( P ) to ( Q ). So, ( ds = |PQ| dlambda ).But I already have ( |PQ| = frac{p (1 + t^2)^2}{2 t^2} ), so ( ds = frac{p (1 + t^2)^2}{2 t^2} dlambda ).But then, the surface area ( S_1 ) is the integral of ( 2 pi r(lambda) ds ), where ( r(lambda) ) is the distance from the point on ( PQ ) to the axis of rotation (the directrix ( l )).Wait, actually, the formula for the surface area of revolution is:( S = 2 pi int_{PQ} r(lambda) ds )Where ( r(lambda) ) is the distance from the curve to the axis of rotation.So, I need to compute:( S_1 = 2 pi int_{0}^{1} r(lambda) |PQ| dlambda )But ( r(lambda) ) is ( x(lambda) + frac{p}{2} ), which we have as:( r(lambda) = frac{p}{2} left[ t^2 + lambda left( frac{1}{t^2} - t^2 right) + 1 right] )Simplify:( r(lambda) = frac{p}{2} left( t^2 + 1 + lambda left( frac{1}{t^2} - t^2 right) right) )Let me denote ( A = t^2 + 1 ) and ( B = frac{1}{t^2} - t^2 ), so:( r(lambda) = frac{p}{2} (A + B lambda) )Then, ( S_1 = 2 pi int_{0}^{1} frac{p}{2} (A + B lambda) cdot frac{p (1 + t^2)^2}{2 t^2} dlambda )Simplify constants:( S_1 = 2 pi cdot frac{p}{2} cdot frac{p (1 + t^2)^2}{2 t^2} int_{0}^{1} (A + B lambda) dlambda )Compute the integral:( int_{0}^{1} (A + B lambda) dlambda = A cdot 1 + frac{B}{2} cdot 1^2 = A + frac{B}{2} )So,( S_1 = 2 pi cdot frac{p}{2} cdot frac{p (1 + t^2)^2}{2 t^2} cdot left( A + frac{B}{2} right) )Substitute back ( A = t^2 + 1 ) and ( B = frac{1}{t^2} - t^2 ):( A + frac{B}{2} = (t^2 + 1) + frac{1}{2} left( frac{1}{t^2} - t^2 right) = t^2 + 1 + frac{1}{2 t^2} - frac{t^2}{2} = frac{t^2}{2} + 1 + frac{1}{2 t^2} )Factor:( frac{t^2}{2} + 1 + frac{1}{2 t^2} = frac{1}{2} left( t^2 + 2 + frac{1}{t^2} right) = frac{1}{2} left( t + frac{1}{t} right)^2 )Because ( (t + 1/t)^2 = t^2 + 2 + 1/t^2 ).So, ( A + frac{B}{2} = frac{1}{2} left( t + frac{1}{t} right)^2 )Therefore, plugging back into ( S_1 ):( S_1 = 2 pi cdot frac{p}{2} cdot frac{p (1 + t^2)^2}{2 t^2} cdot frac{1}{2} left( t + frac{1}{t} right)^2 )Simplify constants:( 2 pi cdot frac{p}{2} = pi p )Then,( pi p cdot frac{p (1 + t^2)^2}{2 t^2} cdot frac{1}{2} left( t + frac{1}{t} right)^2 = pi p cdot frac{p (1 + t^2)^2}{4 t^2} cdot left( t + frac{1}{t} right)^2 )Note that ( (1 + t^2)^2 cdot left( t + frac{1}{t} right)^2 = (1 + t^2)^2 cdot left( frac{t^2 + 1}{t} right)^2 = frac{(1 + t^2)^4}{t^2} )Therefore,( S_1 = pi p cdot frac{p (1 + t^2)^4}{4 t^4} )Simplify:( S_1 = frac{pi p^2 (1 + t^2)^4}{4 t^4} )Okay, so that's ( S_1 ).Now, let's compute ( S_2 ), which is the surface area of the sphere with ( MN ) as its diameter.The surface area of a sphere is ( 4 pi R^2 ), where ( R ) is the radius. Since ( MN ) is the diameter, the radius is ( frac{|MN|}{2} ).From earlier, ( |MN| = p left( t + frac{1}{t} right) ), so the radius ( R ) is ( frac{p}{2} left( t + frac{1}{t} right) ).Therefore, ( S_2 = 4 pi left( frac{p}{2} left( t + frac{1}{t} right) right)^2 = 4 pi cdot frac{p^2}{4} left( t + frac{1}{t} right)^2 = pi p^2 left( t + frac{1}{t} right)^2 )So, ( S_2 = pi p^2 left( t + frac{1}{t} right)^2 )Now, let's compare ( S_1 ) and ( S_2 ):We have:( S_1 = frac{pi p^2 (1 + t^2)^4}{4 t^4} )( S_2 = pi p^2 left( t + frac{1}{t} right)^2 )Let me write both expressions in terms of ( t ):First, note that ( t + frac{1}{t} = frac{t^2 + 1}{t} ), so ( left( t + frac{1}{t} right)^2 = frac{(t^2 + 1)^2}{t^2} )Therefore, ( S_2 = pi p^2 cdot frac{(t^2 + 1)^2}{t^2} )So, ( S_2 = frac{pi p^2 (t^2 + 1)^2}{t^2} )Now, ( S_1 = frac{pi p^2 (1 + t^2)^4}{4 t^4} )So, let's write both:( S_1 = frac{pi p^2 (1 + t^2)^4}{4 t^4} )( S_2 = frac{pi p^2 (1 + t^2)^2}{t^2} )To compare ( S_1 ) and ( S_2 ), let's take the ratio ( frac{S_1}{S_2} ):( frac{S_1}{S_2} = frac{ frac{pi p^2 (1 + t^2)^4}{4 t^4} }{ frac{pi p^2 (1 + t^2)^2}{t^2} } = frac{(1 + t^2)^4}{4 t^4} cdot frac{t^2}{(1 + t^2)^2} = frac{(1 + t^2)^2}{4 t^2} )So,( frac{S_1}{S_2} = frac{(1 + t^2)^2}{4 t^2} )Simplify:( frac{(1 + t^2)^2}{4 t^2} = frac{1 + 2 t^2 + t^4}{4 t^2} = frac{1}{4 t^2} + frac{2 t^2}{4 t^2} + frac{t^4}{4 t^2} = frac{1}{4 t^2} + frac{1}{2} + frac{t^2}{4} )So,( frac{S_1}{S_2} = frac{1}{4 t^2} + frac{1}{2} + frac{t^2}{4} )Let me denote ( u = t^2 ), where ( u > 0 ).Then,( frac{S_1}{S_2} = frac{1}{4 u} + frac{1}{2} + frac{u}{4} )Let me write this as:( frac{S_1}{S_2} = frac{u}{4} + frac{1}{2} + frac{1}{4 u} )Combine terms:( frac{S_1}{S_2} = frac{u}{4} + frac{1}{4 u} + frac{1}{2} )Notice that ( frac{u}{4} + frac{1}{4 u} ) is of the form ( a + frac{1}{a} ), which is minimized when ( a = 1 ), giving a minimum value of 2. Therefore,( frac{u}{4} + frac{1}{4 u} geq 2 cdot sqrt{ frac{u}{4} cdot frac{1}{4 u} } = 2 cdot frac{1}{4} = frac{1}{2} )Wait, actually, the AM-GM inequality states that for positive real numbers ( a ) and ( b ), ( frac{a + b}{2} geq sqrt{ab} ), with equality when ( a = b ).So, applying AM-GM to ( frac{u}{4} ) and ( frac{1}{4 u} ):( frac{ frac{u}{4} + frac{1}{4 u} }{2} geq sqrt{ frac{u}{4} cdot frac{1}{4 u} } = sqrt{ frac{1}{16} } = frac{1}{4} )Therefore,( frac{u}{4} + frac{1}{4 u} geq frac{1}{2} )So,( frac{S_1}{S_2} = frac{u}{4} + frac{1}{4 u} + frac{1}{2} geq frac{1}{2} + frac{1}{2} = 1 )Thus,( frac{S_1}{S_2} geq 1 implies S_1 geq S_2 )Equality holds when ( frac{u}{4} = frac{1}{4 u} implies u^2 = 1 implies u = 1 implies t^2 = 1 implies t = 1 ) (since ( t > 0 ))Therefore, ( S_1 geq S_2 ), with equality when ( t = 1 ).So, the correct conclusion is that ( S_1 geq S_2 ), which corresponds to option C.**Final Answer**boxed{C}