Question

Angle between pair of tangents to a circle formula

Answer 1

Answer:

Find the angle between two tangents from an external point (x1,y1) to the circle x2+y2=a2 .

Below is my attempt at the problem:

x2+y2=a2,Differentiating we get dydx=−xy.

Hence slope will be −a1b1 at the point (a1,b1) for the first tangent.

Similarly slope will be −a2b2 at the point (a2,b2) for the second tangent.

Now a21+b21=a2,a22+b22=a2⟹(a2−a1)(a2+a1)+(b2−b1)(b2+b1)=0

−a1b1=y1−b1x1−a1⟹a1x1+y1b1=a21+b21

Similarly a2x2+y2b2=a22+b22

Now to find the angle between them we have to evaluate tanα=a1b2−a2b1b1b2+a1a2

How to eliminate a1,b1,a2,b2? from here? Please help.