JEE Main 2023 — Parabola Question with Solution

Given parabola $$y^2=24x=4.6x=4ax$$

$$\therefore$$ a = 6

If $$y=mx+c$$ is a tangent to the parabola then,

$$c=\frac{a}{m}$$

$$\Rightarrow c=\frac{6}{m}$$

Let midpoint of chord AB is $$(x_1,y_1)$$

We know, locus of midpoint of chord for hyperbola,

$$T=S_1$$

$$\Rightarrow \frac{x{y}_1+x_{1}y}{2}-2=x_1{y}_1-2$$

$$\Rightarrow x{y}_1+x_{1}y=2x_1{y}_1$$

$$\Rightarrow y=\left(-\frac{y_1}{x_1}\right)x+2y_1$$

This equation is the same tangent to the parabola.

Then comparing with $$y=mx+c$$ we get,

$$c=2y_1,m=-\frac{y_1}{x_1}$$

$$\therefore$$ $$2y_1=\frac{6}{-\frac{y_1}{x_1}}$$

$$\Rightarrow -y_1^2=3x_1$$

$$\therefore$$ Locus of midpoint $$(x_1,y_1)$$ is,

$$-y^2=3x$$

$$\therefore$$ Length of latus rectum = 3

Equation of directrix is $x=\frac{3}{4}$