Question

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

• A. 0
• B. $\pi /3$
• C. $\pi /6$
• D. $\pi /2$

Answer 1

Answer: (D)

Let the equation of tangent $y=mx$ of the circle $(x-1{)}^2+(y+1{)}^2=25$
$\therefore$ Distance from centre of the circle to the tangent = radius of the circle
$\Rightarrow \frac{-1+m}{\sqrt{1+m^2}}=5$
$\Rightarrow 1^2+m^2-2m=25\left(1+m^2\right)$
$\Rightarrow 24m^2+2m+24=0$
$\Rightarrow 12m^2+m+12=0$
$\Rightarrow m_1{m}_2=-1$
Alternate Solution:

It is clear from the figure that circle touches the coordinate axes, therefore tangent is perpendicular.