Question

the length of a chord of a circle $x^2+y^2=9$ intercepted by the line $x+2y=3$ is

• A. $8$
• B. $12\sqrt{5}$
• C. $\frac{12}{\sqrt{5}}$
• D. $5\sqrt{1}2$

Answer 1

Answer: (C)

$\frac{12}{\sqrt{5}}$

The general form of point on the given circle is $(3cos\theta ,3sin\theta )$ and it lies on the line $x+2y=3$

$\Rightarrow cos\theta +2sin\theta =1$

$\Rightarrow 2sin\theta =1-cos\theta$

By squaring on both sides we get $4-4co{s}^2\theta =1+co{s}^2\theta -2cos\theta$

$\Rightarrow 5co{s}^2\theta -2cos\theta -3=0$

$\Rightarrow cos\theta =\frac{2\pm 8}{10}=1,-\frac{3}{5}$

So the end points of chord are $(3,0)$ and $(-\frac{9}{5},\frac{12}{5})$

The distance between them is $\sqrt{\frac{576}{25}+\frac{144}{25}}=\sqrt{\frac{720}{25}}=\sqrt{\frac{144}{5}}=\frac{12}{\sqrt{5}}$

Therefore option $C$ is correct