Question

The angle between the two tangents from the origin to the circle ${(x-7)}^2+{(y+1)}^2=25$ equals-

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{4}$
• D. None of these.

Answer 1

The correct option is A $\frac{\pi }{2}$

Center is $(7,-1)$ and radius $=5$
Let equation of tangent from the origin be $y=mx$ $\Rightarrow mx-y=0$
Then, $\left|\frac{7m+1}{\sqrt{m^2+1}}\right|=5$
$\Rightarrow {(7m+1)}^2=25(m^2+1)\Rightarrow 24m^2+14m-24=0$
Let $m_1$ and $m_2$ be the slopes of the two tangents.
Since $m_1{m}_2=-\frac{24}{24}=-1$
The two tangents are at right angles.