Question

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

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{6}$
• C. $\frac{\pi }{2}$

Answer 1

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

Any line through (0, 0) be y - mx = 0 and it is a tangent to circle $(x-7{)}^2+(y+1{)}^2=25,$ if
$\frac{-1-7m}{\sqrt{1+m^2}}=5\Rightarrow m=\frac{3}{4},-\frac{4}{3}.$
Therefore, the product of both the slopes is –1.
i.e., $\frac{3}{4}\times \frac{4}{3}=-1.$

Hence the angle between the two tangents is $\frac{\pi }{2}.$