Question

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

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{2}$
• D. $\frac{2\pi }{3}$

Answer 1

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

The equation of any line through the origin $(0,0)$ is $y=mx$.
If it is a tangent to the circle $(x-7{)}^2+(y+1{)}^2=5^2$, then
$\left|\frac{7m+1}{\sqrt{m^2+1}}\right|=5$
$\Rightarrow (7m+1=25(m^2+1)\Rightarrow 24m^2+14m-24=0...(i)$
This equation, being a quadratic in $m$, gives two values of $m$, say $m_1$ and $m_2$. These two values of $m$ are the slopes of the tangents drawn from the origin to the given circle.
From $\left(i\right)$, we have $m_1{m}_2=-1$.
Hence, the two tangents are perpendicular.
Ans: C