answer：Alright, so I've got this sequence problem here. It starts with ( a_1 = 1 ) and ( a_2 = 2 ). Then, for each subsequent term, there's this rule depending on whether the previous two terms are odd or even. If both ( a_{n+1} ) and ( a_n ) are odd, then ( a_{n+2} = 5a_{n+1} - 3a_n ). If both are even, then ( a_{n+2} = a_{n+1} - a_n ). Hmm, interesting. I need to figure out something about this sequence, maybe its behavior or a pattern.First, let me write down the initial terms to see if I can spot a pattern. I know ( a_1 = 1 ) and ( a_2 = 2 ). Now, let's compute ( a_3 ). Since ( a_2 = 2 ) is even and ( a_1 = 1 ) is odd, they are not both odd or both even. Wait, the rule only applies if both are odd or both are even. So, does that mean if they are mixed, we don't apply either rule? Or is there a default? The problem statement doesn't specify, so maybe I need to assume that the rule only applies when both are odd or both are even, and otherwise, maybe the sequence doesn't continue? That doesn't make sense because the problem defines ( a_{n+2} ) based on those conditions. Maybe I misread.Wait, looking back, it says ( a_{n+2} ) is defined as either ( 5a_{n+1} - 3a_n ) if both are odd or ( a_{n+1} - a_n ) if both are even. So, if they are mixed, neither condition is met, so does that mean ( a_{n+2} ) isn't defined? That can't be right because the sequence should continue. Maybe I need to clarify.Perhaps the problem assumes that the conditions are exclusive, meaning if both are odd, use the first rule; if both are even, use the second rule; and if one is odd and the other is even, maybe it defaults to one of the rules? But the problem doesn't specify. Hmm, this is confusing. Maybe I should proceed with the assumption that the conditions are exclusive, and if neither condition is met, the sequence doesn't follow either rule, but that seems problematic.Wait, maybe I should check the parity of ( a_2 ) and ( a_1 ). ( a_2 = 2 ) is even, ( a_1 = 1 ) is odd. So, they are mixed. Therefore, neither rule applies. So, does that mean ( a_3 ) isn't defined? That can't be. Maybe I need to re-examine the problem statement.Looking again: "Given ( a_{1}=1, a_{2}=2, a_{n+2}=left{begin{array}{ll}5a_{n+1}-3a_{n} & text{(if } a_{n+1} text{ and } a_{n} text{ are odd)} a_{n+1}-a_{n} & text{(if } a_{n+1} text{ and } a_{n} text{ are even)}end{array}right. )".So, it only defines ( a_{n+2} ) when both ( a_{n+1} ) and ( a_n ) are odd or both are even. If they are mixed, it doesn't define ( a_{n+2} ). That seems like a problem because the sequence would terminate at ( a_2 ). But that can't be right because the problem is asking about the sequence, implying it continues. Maybe I misinterpret the conditions.Wait, perhaps the conditions are not exclusive. Maybe if both are odd, use the first rule; if both are even, use the second rule; and if one is odd and the other is even, use neither? But then the sequence would stop, which doesn't make sense. Alternatively, maybe the problem assumes that the conditions are checked in order, so if both are odd, use the first rule; otherwise, if both are even, use the second rule; and if neither, maybe the sequence doesn't proceed. But again, that would stop the sequence.This is confusing. Maybe I should look for another approach. Perhaps the problem is designed such that after ( a_2 ), the sequence alternates in parity, making the conditions applicable. Let me try computing ( a_3 ) assuming that the rules apply even if the parities are mixed. Wait, but the problem only defines ( a_{n+2} ) based on the parities of ( a_{n+1} ) and ( a_n ). So, if they are mixed, maybe ( a_{n+2} ) isn't defined, which would mean the sequence can't proceed. That seems odd.Alternatively, perhaps I should consider that the problem assumes that the sequence continues regardless, and the rules are applied based on the parities, even if they are mixed. But that contradicts the problem statement. Hmm.Wait, maybe I should proceed by assuming that the sequence continues, and when the parities are mixed, it defaults to one of the rules. Let's try that. Let's say if ( a_{n+1} ) and ( a_n ) are mixed, we use the second rule ( a_{n+1} - a_n ). So, for ( a_3 ), since ( a_2 = 2 ) is even and ( a_1 = 1 ) is odd, we use ( a_{n+1} - a_n ), so ( a_3 = 2 - 1 = 1 ).Now, ( a_3 = 1 ) is odd, ( a_2 = 2 ) is even. Again, mixed. So, ( a_4 = a_3 - a_2 = 1 - 2 = -1 ).( a_4 = -1 ) is odd, ( a_3 = 1 ) is odd. Both odd, so ( a_5 = 5a_4 - 3a_3 = 5*(-1) - 3*1 = -5 - 3 = -8 ).( a_5 = -8 ) is even, ( a_4 = -1 ) is odd. Mixed, so ( a_6 = a_5 - a_4 = -8 - (-1) = -7 ).( a_6 = -7 ) is odd, ( a_5 = -8 ) is even. Mixed, so ( a_7 = a_6 - a_5 = -7 - (-8) = 1 ).( a_7 = 1 ) is odd, ( a_6 = -7 ) is odd. Both odd, so ( a_8 = 5a_7 - 3a_6 = 5*1 - 3*(-7) = 5 + 21 = 26 ).( a_8 = 26 ) is even, ( a_7 = 1 ) is odd. Mixed, so ( a_9 = a_8 - a_7 = 26 - 1 = 25 ).( a_9 = 25 ) is odd, ( a_8 = 26 ) is even. Mixed, so ( a_{10} = a_9 - a_8 = 25 - 26 = -1 ).( a_{10} = -1 ) is odd, ( a_9 = 25 ) is odd. Both odd, so ( a_{11} = 5a_{10} - 3a_9 = 5*(-1) - 3*25 = -5 - 75 = -80 ).( a_{11} = -80 ) is even, ( a_{10} = -1 ) is odd. Mixed, so ( a_{12} = a_{11} - a_{10} = -80 - (-1) = -79 ).( a_{12} = -79 ) is odd, ( a_{11} = -80 ) is even. Mixed, so ( a_{13} = a_{12} - a_{11} = -79 - (-80) = 1 ).Hmm, I'm noticing a pattern here. After ( a_7 = 1 ), the sequence seems to repeat every 6 terms: 1, 26, 25, -1, -80, -79, 1, and so on. Let me check:( a_7 = 1 )( a_8 = 26 )( a_9 = 25 )( a_{10} = -1 )( a_{11} = -80 )( a_{12} = -79 )( a_{13} = 1 )Yes, it repeats every 6 terms after ( a_7 ). So, the sequence becomes periodic with period 6 starting from ( a_7 ). That's interesting.But wait, let me verify the calculations to make sure I didn't make a mistake:- ( a_1 = 1 )- ( a_2 = 2 )- ( a_3 = 2 - 1 = 1 ) (since mixed parity, using second rule)- ( a_4 = 1 - 2 = -1 ) (mixed parity)- ( a_5 = 5*(-1) - 3*1 = -5 - 3 = -8 ) (both odd)- ( a_6 = -8 - (-1) = -7 ) (mixed parity)- ( a_7 = -7 - (-8) = 1 ) (mixed parity)- ( a_8 = 5*1 - 3*(-7) = 5 + 21 = 26 ) (both odd)- ( a_9 = 26 - 1 = 25 ) (mixed parity)- ( a_{10} = 25 - 26 = -1 ) (mixed parity)- ( a_{11} = 5*(-1) - 3*25 = -5 - 75 = -80 ) (both odd)- ( a_{12} = -80 - (-1) = -79 ) (mixed parity)- ( a_{13} = -79 - (-80) = 1 ) (mixed parity)Yes, the calculations seem correct. So, the sequence enters a loop of 1, 26, 25, -1, -80, -79, and then repeats.But wait, the problem didn't specify what to find. It just gave the recurrence relation. Maybe the question is to show that the sequence is periodic or to find its period? Or perhaps to find a closed-form expression?Given that the sequence becomes periodic after a certain point, maybe the question is to prove that the sequence is periodic or to find its period.Alternatively, perhaps the problem is to show that all terms are non-zero, or something like that.Wait, looking back at the initial problem, it just defines the sequence and then says "Please reason step by step, and put your final answer within boxed{}." So, maybe the question is to find a particular term or to show a property of the sequence.Given that I've computed up to ( a_{13} ) and noticed periodicity, maybe the question is to show that the sequence is periodic with period 6 starting from ( a_7 ).Alternatively, perhaps the problem is to find the remainder when each term is divided by 3, as I see in the initial table.Wait, in the initial table, there's a column for ( r_n ), which is the remainder when ( a_n ) is divided by 3. Let me check:- ( a_1 = 1 ), ( r_1 = 1 )- ( a_2 = 2 ), ( r_2 = 2 )- ( a_3 = 1 ), ( r_3 = 1 )- ( a_4 = -1 equiv 2 pmod{3} ), so ( r_4 = 2 )- ( a_5 = -8 equiv 1 pmod{3} ), so ( r_5 = 1 )- ( a_6 = -7 equiv 2 pmod{3} ), so ( r_6 = 2 )- ( a_7 = 1 ), ( r_7 = 1 )- ( a_8 = 26 equiv 2 pmod{3} ), so ( r_8 = 2 )- ( a_9 = 25 equiv 1 pmod{3} ), so ( r_9 = 1 )- ( a_{10} = -1 equiv 2 pmod{3} ), so ( r_{10} = 2 )- ( a_{11} = -80 equiv 1 pmod{3} ), so ( r_{11} = 1 )- ( a_{12} = -79 equiv 2 pmod{3} ), so ( r_{12} = 2 )- ( a_{13} = 1 ), ( r_{13} = 1 )So, the remainders modulo 3 are alternating between 1 and 2 starting from ( a_1 ). That's interesting. So, ( r_n ) alternates 1, 2, 1, 2, etc.This suggests that ( a_n equiv 1 pmod{3} ) when ( n ) is odd and ( a_n equiv 2 pmod{3} ) when ( n ) is even, or vice versa. Wait, looking at the table:- ( a_1 equiv 1 pmod{3} )- ( a_2 equiv 2 pmod{3} )- ( a_3 equiv 1 pmod{3} )- ( a_4 equiv 2 pmod{3} )- ( a_5 equiv 1 pmod{3} )- ( a_6 equiv 2 pmod{3} )- ( a_7 equiv 1 pmod{3} )- ( a_8 equiv 2 pmod{3} )- ( a_9 equiv 1 pmod{3} )- ( a_{10} equiv 2 pmod{3} )- ( a_{11} equiv 1 pmod{3} )- ( a_{12} equiv 2 pmod{3} )- ( a_{13} equiv 1 pmod{3} )So, indeed, the remainders alternate between 1 and 2, starting with 1 for ( n = 1 ). Therefore, for odd ( n ), ( a_n equiv 1 pmod{3} ), and for even ( n ), ( a_n equiv 2 pmod{3} ).This is a useful property. Maybe the problem is to prove this periodicity modulo 3 or to show that all terms are non-zero, which would follow from the fact that ( a_n equiv 1 ) or ( 2 pmod{3} ), hence never 0.Alternatively, perhaps the problem is to find a closed-form expression for ( a_n ), but given the recurrence relation depends on the parity of previous terms, it's non-linear and might not have a straightforward closed-form.Wait, but looking back at the initial problem, it just defines the sequence and then the user instruction is to "Please reason step by step, and put your final answer within boxed{}." So, maybe the question is to find a particular term or to show a property.Given that I've noticed the periodicity modulo 3 and the sequence becoming periodic with period 6, perhaps the problem is to show that the sequence is periodic or to find its period.Alternatively, maybe the problem is to show that ( a_n ) is never zero, which would follow from the fact that ( a_n equiv 1 ) or ( 2 pmod{3} ), hence ( a_n neq 0 ).Wait, in the initial table, the user wrote "Conclusion: From the sequence's behavior modulo 3, for all ( n ), ( a_n neq 0 )." So, perhaps the problem is to show that ( a_n neq 0 ) for all ( n ).Given that, I can structure my thought process as follows:1. Compute initial terms to observe patterns.2. Notice that the sequence alternates between odd and even terms, leading to the application of different recurrence rules.3. Observe that the sequence becomes periodic modulo 3, alternating between 1 and 2.4. Conclude that since ( a_n equiv 1 ) or ( 2 pmod{3} ), ( a_n ) can never be 0.Therefore, the final answer is that ( a_n neq 0 ) for all ( n ).But wait, in my calculations, I saw that ( a_4 = -1 ), which is non-zero, and ( a_5 = -8 ), also non-zero, and so on. So, indeed, all terms are non-zero.Alternatively, maybe the problem is to find the period modulo 3, which is 2, or to find the period of the sequence itself, which seems to be 6 starting from ( a_7 ).But given the initial table and the conclusion that ( a_n neq 0 ), I think the intended answer is that ( a_n ) is never zero.So, to summarize my thought process:- I started by computing the initial terms, noticing the application of different recurrence rules based on parity.- I observed that the sequence alternates between odd and even terms, leading to a periodic behavior modulo 3.- I noticed that the remainders modulo 3 alternate between 1 and 2, implying that ( a_n ) is never divisible by 3, hence ( a_n neq 0 ).Therefore, the conclusion is that ( a_n neq 0 ) for all ( n ).

answer：First, I need to determine the total amount of material Cheryl bought by adding the two quantities she purchased: 3/8 square yards and 1/3 square yards.To add these fractions, I'll find a common denominator. The least common multiple of 8 and 3 is 24.Converting each fraction to have a denominator of 24:- 3/8 becomes 9/24- 1/3 becomes 8/24Adding these together:9/24 + 8/24 = 17/24 square yardsCheryl has 15/40 square yards of material left after the project. To find out how much she used, I'll subtract the leftover material from the total amount bought.First, I'll convert both fractions to have a common denominator. The least common multiple of 24 and 40 is 120.Converting each fraction:- 17/24 becomes 85/120- 15/40 becomes 45/120Subtracting these:85/120 - 45/120 = 40/120Simplifying the result:40/120 = 1/3Therefore, Cheryl used 1/3 square yards of material for her project.

answer：Okay, so I have this geometry problem here. Let me try to visualize it first. There's a circle with a radius of 11 cm, and there's a point P inside the circle that's 7 cm away from the center. A chord of length 18 cm passes through this point P. I need to find the lengths of the two segments that this chord is divided into by point P.Hmm, let me recall some geometry concepts. I remember that when a chord passes through a point inside a circle, there's a theorem related to the products of the segments. I think it's called the Power of a Point theorem. Yeah, that sounds right. It says that for two chords intersecting at a point, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.Wait, in this case, we only have one chord passing through point P. But I think the theorem still applies because if I draw another chord through P, it would intersect the first chord at P, and then the products of the segments would be equal. But since we only have one chord given, maybe I need to create another chord to apply the theorem.Let me think. If I draw a diameter through point P, that would be another chord passing through P. The diameter would pass through the center of the circle, so it would be 22 cm long because the radius is 11 cm. Since point P is 7 cm away from the center, the diameter would be divided into two segments by P: one segment from the center to P, which is 7 cm, and the other from P to the other end of the diameter, which would be 11 cm + 7 cm = 18 cm? Wait, no, that doesn't make sense. The diameter is 22 cm, so if P is 7 cm from the center, then one segment is 7 cm, and the other segment is 22 - 7 = 15 cm. Wait, no, that's not right either because the diameter is 22 cm, so from the center to one end is 11 cm, and from the center to P is 7 cm, so from P to the other end should be 11 cm + 7 cm = 18 cm. Wait, that adds up to 25 cm, which is more than the diameter. Hmm, I must be making a mistake here.Let me clarify. The diameter is 22 cm, so from one end to the other is 22 cm. If the center is O, and P is 7 cm away from O, then the distance from P to one end of the diameter would be 11 cm - 7 cm = 4 cm, and from P to the other end would be 11 cm + 7 cm = 18 cm. Yes, that makes sense because 4 cm + 18 cm = 22 cm, which is the length of the diameter. Okay, so the diameter is divided into segments of 4 cm and 18 cm by point P.Now, applying the Power of a Point theorem, which states that for two chords intersecting at P, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord. In this case, the two chords are the given chord AB of length 18 cm and the diameter CD. So, if I let AP = x and PB = 18 - x, then according to the theorem:AP * PB = CP * PDWhere CP and PD are the segments of the diameter CD. From earlier, CP is 18 cm and PD is 4 cm. So,x * (18 - x) = 18 * 4Let me compute the right side first:18 * 4 = 72So,x * (18 - x) = 72Expanding the left side:18x - x² = 72Rearranging the equation:x² - 18x + 72 = 0Now, I need to solve this quadratic equation. Let me try factoring it. I'm looking for two numbers that multiply to 72 and add up to -18. Hmm, let's see:- The factors of 72 are: 1 & 72, 2 & 36, 3 & 24, 4 & 18, 6 & 12, 8 & 9.Looking for a pair that adds up to 18. Wait, 6 and 12 multiply to 72 and add up to 18. But since the middle term is -18x, both numbers should be negative to add up to -18 and multiply to 72. So, -6 and -12.Therefore, the equation factors as:(x - 6)(x - 12) = 0Setting each factor equal to zero:x - 6 = 0 => x = 6x - 12 = 0 => x = 12So, the solutions are x = 6 cm and x = 12 cm. Therefore, the lengths of the segments AP and PB are 6 cm and 12 cm, respectively.Wait, let me double-check. If AP is 6 cm, then PB would be 18 - 6 = 12 cm. And according to the Power of a Point theorem, AP * PB should equal CP * PD, which is 18 * 4 = 72. Let's verify:6 * 12 = 72Yes, that's correct. So, the calculations seem to check out.I think that's the solution. The chord is divided into segments of 6 cm and 12 cm by point P.

answer：Okay, so I have this problem about converting a polar equation to a rectangular one and then finding some value involving distances from a point to the intersection points of a line and the curve. Let me try to break this down step by step.First, part (I) asks me to convert the polar equation ρ = sinθ / cos²θ into a rectangular coordinate equation. Hmm, I remember that in polar coordinates, ρ is the radius, θ is the angle, and the conversion formulas are x = ρ cosθ and y = ρ sinθ. So maybe I can express everything in terms of x and y.Given ρ = sinθ / cos²θ, I can multiply both sides by cos²θ to get ρ cos²θ = sinθ. Now, I know that ρ cosθ is x, so ρ cos²θ would be x * cosθ. Wait, but cosθ is x / ρ, right? So substituting that in, ρ cos²θ = x * (x / ρ) = x² / ρ. So, the left side is x² / ρ, and the right side is sinθ, which is y / ρ.So, putting it together: x² / ρ = y / ρ. Since both sides have 1/ρ, I can multiply both sides by ρ to eliminate the denominators, which gives me x² = y. So, the rectangular equation is y = x². That seems straightforward. I think that's correct because a parabola y = x² is a common curve, and the polar equation given should correspond to that.Moving on to part (II). I need to draw a line l passing through point P(0, 2) with a slope of 1. So, the equation of the line should be y = x + 2, since the slope is 1 and it passes through (0, 2). Let me confirm that: yes, when x = 0, y = 2, and the slope is 1, so the equation is correct.Now, I need to find the points A and B where this line intersects the curve C, which is y = x². So, I can set x² = x + 2. That gives me the quadratic equation x² - x - 2 = 0. Let me solve this quadratic. The discriminant is b² - 4ac = (-1)² - 4*1*(-2) = 1 + 8 = 9. So, the roots are [1 ± √9]/2 = [1 ± 3]/2. Therefore, x = (1 + 3)/2 = 2 and x = (1 - 3)/2 = -1. So, the x-coordinates of points A and B are 2 and -1.To find the corresponding y-coordinates, plug these back into y = x + 2. For x = 2, y = 4, so point A is (2, 4). For x = -1, y = 1, so point B is (-1, 1). So, points A(2, 4) and B(-1, 1).Now, I need to find |PA| and |PB|, where P is (0, 2). Let's compute the distances.First, |PA|: distance between (0, 2) and (2, 4). Using the distance formula, sqrt[(2 - 0)² + (4 - 2)²] = sqrt[4 + 4] = sqrt[8] = 2√2.Next, |PB|: distance between (0, 2) and (-1, 1). Using the distance formula, sqrt[(-1 - 0)² + (1 - 2)²] = sqrt[1 + 1] = sqrt[2].So, |PA| = 2√2 and |PB| = √2.Now, the problem asks for 1/|PA| + 1/|PB|. Let's compute that:1/(2√2) + 1/√2. Let me combine these terms. Since 1/√2 is equal to √2/2, and 1/(2√2) is equal to √2/4. So, adding them together: √2/4 + √2/2. To add these, I need a common denominator, which is 4. So, √2/4 + 2√2/4 = 3√2/4.Wait, that seems too straightforward. Let me double-check my steps.1. Converted the polar equation correctly to y = x². That seems right.2. Equation of the line: y = x + 2. Correct, since slope is 1 and passes through (0,2).3. Intersection points: solved x² = x + 2, which gives x² - x - 2 = 0. Factored or solved using quadratic formula? I used the quadratic formula and got x = 2 and x = -1. Plugging back in, y = 4 and y = 1. So points A(2,4) and B(-1,1). That seems correct.4. Distances:- PA: sqrt[(2)^2 + (2)^2] = sqrt[8] = 2√2. Correct.- PB: sqrt[(-1)^2 + (-1)^2] = sqrt[2]. Correct.5. Reciprocals: 1/(2√2) + 1/√2. Convert to like terms:1/(2√2) = √2/4 and 1/√2 = √2/2 = 2√2/4. So, adding them: √2/4 + 2√2/4 = 3√2/4. That seems correct.Hmm, so the value is 3√2/4. Is there another way to approach this problem, maybe using parametric equations or something else?Wait, in the initial solution, they used parametric equations for the line. Let me see if that approach would yield the same result.Parametrizing the line l passing through P(0,2) with slope 1. So, direction vector is (1,1). So, parametric equations can be written as x = t, y = 2 + t, where t is a parameter. Alternatively, sometimes people use a different parameter, say s, but it's similar.But in the initial solution, they used x = (√2/2)t and y = 2 + (√2/2)t. Hmm, that seems like they're using a different parameterization, perhaps with a different scaling factor. Let me see why.If you use x = t, y = 2 + t, then substituting into y = x² gives t² = 2 + t, which is t² - t - 2 = 0, same as before, solutions t = 2 and t = -1. So, points (2,4) and (-1,1), same as before.Alternatively, if you use x = (√2/2)t, then y = 2 + (√2/2)t. Then, substituting into y = x², we get 2 + (√2/2)t = ( (√2/2)t )² = (2/4)t² = (1/2)t². So, equation becomes (1/2)t² - (√2/2)t - 2 = 0. Multiply both sides by 2: t² - √2 t - 4 = 0. So, discriminant is (√2)^2 - 4*1*(-4) = 2 + 16 = 18. So, roots are [√2 ± √18]/2 = [√2 ± 3√2]/2. So, t = (1 + 3)/2 √2 = 2√2 and t = (1 - 3)/2 √2 = -√2. Wait, that's different from before.Wait, hold on, if t = 2√2, then x = (√2/2)*(2√2) = (√2 * 2√2)/2 = (2*2)/2 = 2. Similarly, y = 2 + (√2/2)*(2√2) = 2 + (2*2)/2 = 2 + 2 = 4. So, point A is (2,4). Similarly, t = -√2, x = (√2/2)*(-√2) = (-2)/2 = -1, y = 2 + (√2/2)*(-√2) = 2 - (2)/2 = 2 - 1 = 1. So, point B is (-1,1). So, same points as before.But in this parametrization, the parameter t is different. So, in this case, t1 = 2√2 and t2 = -√2. So, |PA| and |PB| can be found using the parameter t, since in parametric equations, the parameter often relates to the distance from the point P.Wait, in the parametric equations, x = (√2/2)t and y = 2 + (√2/2)t. So, the parameter t is scaled by √2/2. So, the actual distance from P(0,2) to a point on the line is |t| times the scaling factor. Wait, no, actually, the distance would be sqrt[( (√2/2 t)^2 + ( (√2/2 t)^2 )] = sqrt[ ( (2/4)t² + (2/4)t² ) ] = sqrt[ (4/4)t² ] = sqrt[t²] = |t|. Wait, that can't be right because the scaling factor is √2/2, so the actual distance should be |t| * (√2/2). Wait, no, let me think.Wait, the parametric equations are x = (√2/2)t and y = 2 + (√2/2)t. So, the direction vector is (√2/2, √2/2). The length of this direction vector is sqrt[ (√2/2)^2 + (√2/2)^2 ] = sqrt[ (2/4) + (2/4) ] = sqrt[1] = 1. So, it's a unit vector. Therefore, the parameter t represents the actual distance from the point P(0,2). So, when t = 2√2, the distance PA is 2√2, and when t = -√2, the distance PB is √2. Wait, but in the parametrization, t is the actual distance? Hmm, because the direction vector is a unit vector, so yes, t is the actual distance along the line from P.Wait, but in the initial solution, they computed |PA| + |PB| as |t1| + |t2|, which is |2√2| + | -√2| = 2√2 + √2 = 3√2. Then, |PA||PB| is |t1 t2| = |2√2 * (-√2)| = | -4 | = 4. So, 1/|PA| + 1/|PB| = (|PA| + |PB|) / (|PA||PB|) = (3√2)/4, which is the same as before.So, whether I compute the distances directly using coordinates or use the parameter t, I get the same result. So, both methods confirm that 1/|PA| + 1/|PB| is 3√2 / 4.But wait, in the initial solution, they used the parametric equations and got t1 + t2 = √2 and t1 t2 = -4. Then, they computed |t1| + |t2| as sqrt[(t1 + t2)^2 - 4 t1 t2] = sqrt[(√2)^2 - 4*(-4)] = sqrt[2 + 16] = sqrt[18] = 3√2. Then, |t1 t2| = 4, so 3√2 / 4. That's another way to compute |t1| + |t2| without finding t1 and t2 individually.But in my approach, I found t1 and t2 as 2√2 and -√2, so |t1| + |t2| = 2√2 + √2 = 3√2, same as before. So, both methods work.I think I've covered all the steps, double-checked the calculations, and both approaches lead to the same result. So, I feel confident that the answer is 3√2 / 4.

answer：Okay, so I have this problem where a random variable ξ follows a binomial distribution B(n, P). They've given me the expected value Eξ = 15 and the variance Dξ = 11.25. I need to find the value of n from the options provided: 60, 55, 50, or 45.First, I remember that for a binomial distribution, the expected value Eξ is given by n times the probability of success P, so Eξ = nP. They've told me that this equals 15, so I can write that as:nP = 15 ...(1)Next, the variance of a binomial distribution is given by nP(1 - P). They've given me that the variance Dξ is 11.25, so:nP(1 - P) = 11.25 ...(2)Now, I have two equations here. The first one is nP = 15, and the second one is nP(1 - P) = 11.25. I need to solve these two equations to find the value of n.Looking at equation (1), I can solve for P in terms of n. If nP = 15, then P = 15/n. That makes sense because P is the probability of success, and it should be a value between 0 and 1, so n must be greater than 15.Now, I can substitute this expression for P into equation (2). So, replacing P with 15/n in equation (2), we get:n * (15/n) * (1 - 15/n) = 11.25Simplifying that, the n in the numerator and denominator cancels out:15 * (1 - 15/n) = 11.25Now, I can divide both sides by 15 to make it simpler:(1 - 15/n) = 11.25 / 15Calculating 11.25 divided by 15, I get:11.25 / 15 = 0.75So, now the equation is:1 - 15/n = 0.75Subtracting 1 from both sides:-15/n = 0.75 - 1Which simplifies to:-15/n = -0.25Multiplying both sides by -1 to eliminate the negative signs:15/n = 0.25Now, I can solve for n by taking the reciprocal of both sides:n/15 = 1 / 0.25Calculating 1 divided by 0.25, I get:1 / 0.25 = 4So, n/15 = 4Multiplying both sides by 15:n = 4 * 15n = 60So, the value of n is 60. Looking back at the options provided, 60 is option A. Therefore, the correct answer should be A: 60.Wait, let me double-check my calculations to make sure I didn't make any mistakes. Starting from the beginning:Eξ = nP = 15Var(ξ) = nP(1 - P) = 11.25From Eξ, P = 15/nSubstituting into Var(ξ):n * (15/n) * (1 - 15/n) = 11.25Simplifies to:15 * (1 - 15/n) = 11.25Divide both sides by 15:1 - 15/n = 0.75Subtract 1:-15/n = -0.25Multiply by -1:15/n = 0.25Take reciprocal:n/15 = 4Multiply by 15:n = 60Yes, that seems correct. So, n is indeed 60. Therefore, the correct choice is A: 60.

answer：First, I need to convert the hexadecimal number E78 to its decimal equivalent. In hexadecimal, E represents 14, 7 represents 7, and 8 represents 8. Using the place values of powers of 16, the conversion is as follows:E78 (hex) = (14 * 16²) + (7 * 16¹) + (8 * 16⁰)E78 (hex) = (14 * 256) + (7 * 16) + (8 * 1)E78 (hex) = 3584 + 112 + 8E78 (hex) = 3704 (decimal)Next, I will convert the decimal number 3704 to radix 7. This involves dividing the number by 7 and keeping track of the remainders:3704 ÷ 7 = 529 remainder 1529 ÷ 7 = 75 remainder 475 ÷ 7 = 10 remainder 510 ÷ 7 = 1 remainder 31 ÷ 7 = 0 remainder 1Reading the remainders in reverse order gives the radix 7 number: 13541.Now, I will convert the radix 7 number 13541 back to decimal to verify its equivalence to the original decimal value:13541 (radix 7) = (1 * 7⁴) + (3 * 7³) + (5 * 7²) + (4 * 7¹) + (1 * 7⁰)13541 (radix 7) = (1 * 2401) + (3 * 343) + (5 * 49) + (4 * 7) + (1 * 1)13541 (radix 7) = 2401 + 1029 + 245 + 28 + 113541 (radix 7) = 3704 (decimal)Finally, I will convert the decimal number 3704 back to hexadecimal:3704 ÷ 16 = 231 remainder 8231 ÷ 16 = 14 remainder 714 ÷ 16 = 0 remainder 14 (which is E in hexadecimal)So, the hexadecimal representation of 3704 is E78.The sum of the initial hexadecimal number E78 and the final hexadecimal number E78 is:E78 (hex) + E78 (hex) = 3704 (decimal) + 3704 (decimal) = 7408 (decimal)Now, I will convert 7408 to hexadecimal:7408 ÷ 16 = 463 remainder 0463 ÷ 16 = 28 remainder 15 (which is F in hexadecimal)28 ÷ 16 = 1 remainder 12 (which is C in hexadecimal)1 ÷ 16 = 0 remainder 1So, the hexadecimal representation of 7408 is 1CF0.Therefore, the sum of the initial hexadecimal number E78 and the final hexadecimal number E78 is 1CF0 in hexadecimal.