Question

The angle between the chords of the circle $x^2+y^2=100,$ which passes through the point $\left(7,1\right)$ and also divides the circumference of the circle into two arcs whose lengths are in the ratio $2:1,$ is equal to

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{2}$
• D. $\frac{2\pi }{3}$

Answer 1

Answer: (C)

Let the chord is $AB$ which subtends an angle $\theta$ at the centre $\left(0,0\right)$
$\Rightarrow \theta +2\theta =360^o$
$\Rightarrow \theta =120^o=\angle AOB$

Let the distance of $O$ from $AB=h$
Then, $cos60^o=\frac{h}{10}=\frac{1}{2}\Rightarrow h=5$
Let the equation of the chord is $\frac{y-1}{x-7}=m$
$\Rightarrow mx-y+1-7m=0$ whose distance from $\left(0,0\right)$ is equal to $5$
$\Rightarrow \left|\frac{0-0+1-7m}{\sqrt{1+m^2}}\right|=5$
$\Rightarrow 1-14m+49m^2=25+25m^2$
$\Rightarrow 24m^2-14m-24=0\Rightarrow m_1{m}_2=-1$
$\Rightarrow$ Chords are perpendicular