JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

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

Choose the correct answer:

Explanation

Solution

Tangent to Parabola: $y^2 = 24x$ ki tangent $y = mx + \frac{6}{m}$ hoti hai.
Chord with Midpoint $(h, k)$ : Hyperbola $xy = 2$ ke liye chord ka equation $T = S_1$ se:

$\frac{1}{2}(xk + yh) = hk \implies xk + yh = 2hk$ .
Comparison: Dono equations ek hi line ko darshate hain.

$m = -\frac{k}{h}$ aur $\frac{6}{m} = 2k \implies \frac{6}{-k/h} = 2k \implies -6h = 2k^2$ .
Locus: $k^2 = -3h \implies y^2 = -3x$ .
Parabola Properties: Ise $y^2 = -4ax$ se compare karne par, $4a = 3$ , toh $a = 3/4$ .
- Latus rectum $= 4a = 3$ .
- Directrix: $x = a \implies x = 3/4 \implies 4x = 3$ .

Sahi Option: (D)