Question

Chord $A B$ of the circle $x^{2}+y^{2}=100$ passes through the point $(7,1)$ and subtends an angle of $60^{\circ}$ at the circumference of the circle. If $m_{1}$ and $m_{2}$ are the slopes of two such chords then the value of $m_{1} m_{2}$ is

• A. $-1$
• B. $1$
• C. $7 / 12$
• D. $-3$

Answer 1

Answer: (A)

Let the slope of the chord through point $(7,1)$ be $m$.
Thus, equation of line is
$y-1=m(x-7) $ or $ m x-y+1-7 m=0$
Perpendicular distance from $(0,0)=\frac{r}{2}$
$\Rightarrow \frac{|7 m-1|}{\sqrt{1+m^{2}}}=5 $
$\Rightarrow (7 m-1)^{2}=25\left(1+m^{2}\right) $
$\Rightarrow 49 m^{2}-14 m+1=25+25 m^{2}$
$\Rightarrow 24 m^{2}-14 m-24=0 $
$\Rightarrow m_{1} m_{2}=-1$