If a line along a chord of the circle

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 :

  1. 5

  2. 7

  3. $5 \sqrt{3}$

  4. $3 \sqrt{5}$


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 $m=\frac{1}{2}: y=\frac{x}{2}+15 \Rightarrow x-2 y+30=0$

$\ell_{\mathrm{AB}}=2 \sqrt{\mathrm{R}^{2}-\mathrm{P}^{2}}=2 \sqrt{\frac{225}{4}-\frac{225}{5}}$

$\Rightarrow \ell_{\mathrm{AB}}=30 \cdot \sqrt{\frac{1}{20}}=\frac{15}{\sqrt{5}}=3 \sqrt{5}$

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