A and B are focus of hyperbola 25x−12+y+12=k2 3x+4y+52, k>1 and P is a variable point on hyperbola such that ΔAPB is right angled isosceles triangle. If k=tanα, α∈0, 2π and sum of the values of α is aπ2b (where a and b are co-prime numbers), then which of the following options is/are correct?

Where AP = 2a + 2ae⇒AP2=AB2+PB2⇒4a2+4a2e2+8a2e=4a2e2+4a2e2⇒e2−2e−1=0⇒e=2+222 or 2−222 reject⇒e=1±2⇒k=1+2sum of all values =7π4 a=7 and b=2

A and B are focus of hyperbola 25x−12+y+12=k2 3x+4y+52, k>1 and P is a variable point on hyperbola such that ΔAPB is right angled isosceles triangle. If k=tanα, α∈0, 2π and sum of the values of α is aπ2b (where a and b are co-prime numbers), then which of the following options is/are correct?

Where AP = 2a + 2ae⇒AP2=AB2+PB2⇒4a2+4a2e2+8a2e=4a2e2+4a2e2⇒e2−2e−1=0⇒e=2+222 or 2−222 reject⇒e=1±2⇒k=1+2sum of all values =7π4 a=7 and b=2

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A and B are focus of hyperbola $25 [{(x - 1)}^{2} + {(y + 1)}^{2}] = k^{2} {(3 x + 4 y + 5)}^{2}, k > 1$ and P is a variable point on hyperbola such that $Δ A P B$ is right angled isosceles triangle. If $k = \tan α, α \in [0, 2 π]$ and sum of the values of $α$ is $(\frac{a π}{2 b})$ (where a and b are co-prime numbers), then which of the following options is/are correct?

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a = 6

b = 3

a + b = 9

Sum of all values of $α = \frac{7 π}{4}$

answer is C, D.

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Detailed Solution

Where AP = 2a + 2ae

$\Rightarrow A P^{2} = A B^{2} + P B^{2}$

$\Rightarrow 4 a^{2} + 4 a^{2} e^{2} + 8 a^{2} e = 4 a^{2} e^{2} + 4 a^{2} e^{2}$

$\Rightarrow e^{2} - 2 e - 1 = 0$

$\Rightarrow e = \frac{2 + 2 \sqrt{2}}{2} o r \frac{2 - 2 \sqrt{2}}{2} r e j e c t$

$\Rightarrow e = 1 \pm \sqrt{2} \Rightarrow k = 1 + \sqrt{2}$

$s u m o f a l l v a l u e s = \frac{7 π}{4} a = 7 a n d b = 2$

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