question_answer

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)         \[\pi /2\]

Correct Answer: D

Solution :

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 \[\Rightarrow \]               \[\frac{-1+m}{\sqrt{1+{{m}^{2}}}}=5\] \[\Rightarrow \]               \[{{1}^{2}}+{{m}^{2}}-2m=25(1+{{m}^{2}})\] \[\Rightarrow \]               \[24{{m}^{2}}+2m+24=0\] \[\Rightarrow \]               \[12\,{{m}^{2}}+m+12=0\] \[\Rightarrow \]               \[{{m}_{1}}{{m}_{2}}=-1\] Alternate Solution: It is clear from the figure that circle touches the coordinate axes, therefore tangent is perpendicular.