JEE Main 2021 — Parabola Question with Solution

We have,

$4x^2+4y^2+120x+675=0$

$\Rightarrow x^2+y^2+30x+\frac{675}{4}=0$

Centre$\equiv \left(-15,0\right)$ and radius$=\sqrt{225-\frac{675}{4}}=\frac{15}{2} \mathrm{units}$

Let the chord cuts the circle at $A \& B$ and passes through $P\left(-30,0\right)$

Equation of tangent to parabola $y^2=30x$ is

$y=mx+\frac{30}{4m} ...\left(\mathrm{i}\right)$

It passes through $\left(-30,0\right)$, so

$0=-30m+\frac{30}{4m}$

$\Rightarrow 4m^2=1$

$\Rightarrow m=\pm \frac{1}{2}$

For $m=\frac{1}{2}$, we have

$y=\frac{x}{2}+15$

$\Rightarrow x-2y+30=0$

Length of $OB=\frac{-15+0+30}{\sqrt{5}}=3\sqrt{5}$

Length of $BC=\sqrt{\frac{225}{4}-45}\Rightarrow BC=\frac{3\sqrt{5}}{2}$

Length of chord $AC=2\times \frac{3\sqrt{5}}{2}=3\sqrt{5}$.

Similarly for $m=-\frac{1}{2}$, length of chord $AC=3\sqrt{5}$.