Question:
If a line along a chord of the circle $4 x^{2}+4 y^{2}+120 x+675=0$, passes through the point $(-30,0)$ and is tangent to the parabola $y^{2}=30 x$, then the length of this chord is :
Correct Option: , 4
Solution:
Equation of tangent to $\mathrm{y}^{2}=30 \mathrm{x}$
$\mathrm{y}=\mathrm{m} \mathrm{x}+\frac{30}{4 \mathrm{~m}}$
Pass thru $(-30,0): a=-30 m+\frac{30}{4 m} \Rightarrow m^{2}=1 / 4$
$\Rightarrow \mathrm{m}=\frac{1}{2}$ or $\mathrm{m}=-\frac{1}{2}$
At $m=\frac{1}{2}: y=\frac{x}{2}+15 \Rightarrow x-2 y+30=0$
$\ell_{\mathrm{AB}}=2 \sqrt{\mathrm{R}^{2}-\mathrm{P}^{2}}=2 \sqrt{\frac{225}{4}-\frac{225}{5}}$
$\Rightarrow \ell_{\mathrm{AB}}=30 \cdot \sqrt{\frac{1}{20}}=\frac{15}{\sqrt{5}}=3 \sqrt{5}$