Consider the circle and the parabola . If the set of all values of , for which three chords of the circle C on three distinct lines passing through the point are bisected by the parabola P is interval (p, q), then is equal to __________. [2024

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Solution

Q.
Consider the circle $C : x^{2} + y^{2} = 4$ and the parabola $P : y^{2} = 8 x$ . If the set of all values of $α$ , for which three chords of the circle C on three distinct lines passing through the point $(α, 0)$ are bisected by the parabola P is the interval (p, q), then ${(2 q - p)}^{2}$ is equal to __________. [2024]

Ans.
(80)

Equation of chord of parabola whose mid-point is $(α, 0)$ is given by $T = S_{1}$

$\Rightarrow y y_{1} - 4 (x + x_{1}) = y_{1}^{2} - 8 x_{1}$

$i . e ., - 4 (x + α) = 0 - 8 α$

$\Rightarrow x = α$

Equation of chord of circle with $A (2 t^{2}, 4 t)$ as mid-point is

$x x_{1} + y y_{1} - 4 = x_{1}^{2} + y_{1}^{2} - 4$

$\Rightarrow 2 t^{2} x + 4 t y = 4 t^{4} + 16 t^{2}$

Also, it passes through $(α, 0)$

$\Rightarrow 2 t^{2} α = 4 t^{4} + 16 t^{2}$

$\Rightarrow 2 α = 4 t^{2} + 16$

$\Rightarrow α = 2 t^{2} + 8 = x_{0} + 8$

Also, we have $x^{2} + y^{2} = 4$ and $y^{2} = 8 x$

$\Rightarrow x^{2} + 8 x - 4 = 0$

$\Rightarrow x_{0} = \frac{- 8 + \sqrt{80}}{2}$

$\Rightarrow p = 8 and q = 4 + 2 \sqrt{5}$

$∴ {(2 q - p)}^{2} = 80$ .

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