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

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