Two parabolas have the same focus (4, 3) and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersects at the points A and B, then is equal to : [2025]

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Solution

Q.
Two parabolas have the same focus (4, 3) and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersects at the points A and B, then ${(A B)}^{2}$ is equal to : [2025]

1 392

2 96

3 384

4 192

Ans.
(4)

Let the two parabolas intersect at $A (x_{1}, y_{1})$ and $B (x_{2}, y_{2})$ .

$∴$ Equation of parabolas are

${(x - 4)}^{2} + {(y - 3)}^{2} = x^{2}$ ... (i)

and ${(x - 4)}^{2} + {(y - 3)}^{2} = y^{2}$ ... (ii)

From (i) and (ii), we get x = y

$\Rightarrow x^{2} - 14 x + 25 = 0$ and

$\Rightarrow x_{1} + x_{2} = 14 and x_{1} x_{2} = 25$

$∴ A B^{2} = {(x_{1} - x_{2})}^{2} + {(y_{1} - y_{2})}^{2} = 2 {(x_{1} - x_{2})}^{2}$

$= 2 {[(x_{1} + x_{2})}^{2} - 4 x_{1} x_{2}] = 2 [14^{2} - 100] = 192$ .

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