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

(A) directrix $4 \mathrm{x}=3$

(B) directrix $4 \mathrm{x}=-3$

(C) Length of latus rectum $\frac{3}{2}$

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Parabola

Let a tangent to the curve

y^{2} = 24 x

meet the curve

xy = 2

at the points A and B . Then the mid points of such line segments AB lie on a parabola with the

(A)directrix $4 x = 3$ ✓ Correct
(B)directrix $4 x = - 3$
(C)Length of latus rectum $\frac{3}{2}$
(D)Length of latus rectum 2

Answer: (A)

Solution

A tangent to

y^{2} = 24 x

t y = x + 6 t^{2}

. For the rectangular hyperbola

x y = 2

, the chord bisected at

(h, k)

k x + h y = 2 hk

. Matching the coefficients of this chord with the tangent gives

h = - t k

and

h = - 3 t^{2}

, so

k = 3 t

and

h = - \frac{k ^{2}}{3}

; the midpoints therefore satisfy

y^{2} = - 3 x

. This parabola has

4 a = 3

, placing its directrix at

x = \frac{3}{4}

, that is

4 x = 3.

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Single correct option

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