JEE MAIN - Mathematics (2024 - 4th April Evening Shift - No. 13)
Let $$P Q$$ be a chord of the parabola $$y^2=12 x$$ and the midpoint of $$P Q$$ be at $$(4,1)$$. Then, which of the following point lies on the line passing through the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ ?
$$(3,-3)$$
$$\left(\frac{1}{2},-20\right)$$
$$(2,-9)$$
$$\left(\frac{3}{2},-16\right)$$
Explanation
$$y^2=12 x$$
Chord $$P Q$$ having mid-point $$(x_1, y_1)=(4,1)$$ equation of chord $$P Q$$
$$\begin{aligned} & T=S_1 \\ & y y_1-12 \frac{\left(x+x_1\right)}{2}=y_1^2-12 x_1 \\ & y-6(x+4)=1-12 \times 4 \\ & y-6 x-24=-47 \\ & y-6 x+23=0 \end{aligned}$$
From option (4) $$x=\frac{1}{2}$$ & $$y=-20$$
$$-20-6 \times \frac{1}{2}+23=0$$
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