Question

Chord $AB$ of the circle $x^2+y^2=100$ passes through the point $(7,1)$ and subtends an angle of ${60}^o$ 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

The correct option is A $-1$

Let the slope of the chord through $(7,1)$ be 4 $m$

Equation of line is

$\Rightarrow y-1=m(x-7)$

$\Rightarrow mx-y+(1-7m)=0$

From the figure $\Rightarrow \cos {60}^0=\frac{OM}{OA}$

$\Rightarrow \frac{1}{2}=\frac{OM}{r}$

$\Rightarrow OM=\frac{r}{2}=\frac{10}{2}=5$

Perpendicular distance from $O(0,0)$ to $M=\frac{r}{2}$

$\Rightarrow \frac{|7m-1|}{\sqrt{1+m^2}}=5$

$\Rightarrow (7m-1{)}^2=25(1+m^2)$

$\Rightarrow 24m^2-14m-24=0$

Let $m_1$ , $m_2$ be the roots of the equation

Then product of the roots $m_1{m}_2=\frac{-24}{24}=-1$

$\Rightarrow m_1{m}_2=-1$