Let a tangent to the curve y 2 = 24 x meet the curve x y = 2 at the points A and B. Then the mid-points of such line segments AB lie on a parabola with the [2023]

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Solution

Q.
Let a tangent to the curve $y^{2} = 24 x$ meet the curve $x y = 2$ at the points A and B. Then the mid-points of such line segments AB lie on a parabola with the [2023]

1 length of latus rectum 3/2

2 length of latus rectum 2

3 directrix $4 x$ = 3

4 directrix $4 x$ = - 3

Ans.
(3)

$Let the equation of tangent to y^{2} = 24 x$ $is t y = x + 6 t^{2}$

$Now, t y = x + 6 t^{2} meets the curve x y = 2 at points A and B .$

$Let mid-point of A B be (h, k)$

$t y = \frac{2}{y} + 6 t^{2}$ $[∵ x = \frac{2}{y}]$

$t y^{2} - 6 t^{2} y - 2 = 0$

$y_{1} + y_{2} = 6 t$

$t \cdot \frac{2}{x} = x + 6 t^{2}$ $[∵ y = \frac{2}{x}]$

$x^{2} + 6 t^{2} x - 2 t = 0$

$x_{1} + x_{2} = - 6 t^{2}$

$Midpoint P is (- 3 t^{2}, 3 t)$

$\Rightarrow h = - 3 t^{2}, k = 3 t$ $\Rightarrow (\frac{h}{- 3}) = {(\frac{k}{3})}^{2} \Rightarrow y^{2} = - 3 x$

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