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, $cos 60^{o}=\frac{h}{10}=\frac{1}{2}\Rightarrow h=5$
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 $5$
$\Rightarrow \left|\frac{0 - 0 + 1 - 7 m}{\sqrt{1 + m^{2}}}\right|=5$
$\Rightarrow 1-14m+49m^{2}=25+25m^{2}$
$\Rightarrow 24m^{2}-14m-24=0\Rightarrow m_{1}m_{2}=-1$
$\Rightarrow $ Chords are perpendicular