Let be a point on the parabola . If P also lies on the chord of the parabola whose mid point is , then is equal to __________. [2024]

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Solution

Q.
Let $P (α, β)$ be a point on the parabola $y^{2} = 4 x$ . If P also lies on the chord of the parabola $x^{2} = 8 y$ whose mid point is $(1, \frac{5}{4})$ , then $(α - 28) (β - 8)$ is equal to __________. [2024]

Ans.
(192)

We $α = t^{2}, β = 2 t$

Now, equation of chord of the parabola $x^{2} = 8 y$ , bisected at $(1, \frac{5}{4})$ is $x \cdot 1 - 2 \times 2 (y + \frac{5}{4}) = 1 - 10$

$\Rightarrow x - 4 y - 5 = - 9 \Rightarrow x - 4 y + 4 = 0$

Now, $(α, β)$ satisfies the above equation

$∴ t^{2} - 8 t + 4 = 0$ ... (i)

Now, $(α - 28) (β - 8) = (t^{2} - 28) (2 t - 8)$

$= (8 t - 4 - 28) (2 t - 8)$ (Using (i))

$= 16 t^{2} - 64 t - 64 t + 256 = 16 (t^{2} - 8 t + 16)$

$= 16 (t^{2} - 8 t + 4 + 12) = 192$ .

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