Question

If $\theta$ is the angle between the circles passing through $\left(\lambda,2\lambda\right),(2\lambda,\lambda)$ and touching x-axis, then the locus of the point from which pair of tangents are drawn to the circle $x^2+y^2-2x-2y-7=0$ which includes the angle $\theta$ is

Answer 1

Answer:

The equations of the circles are:

x

2

+y

2

−2λx−2y−7=0

Hence, g=−λ,f=−1,c=−7 ....(1)

3x

2

+3y

2

−8x+29y=0

Hence, g

′

=−

3

4

,f

′

=

6

29

,c

′

=0 ....(2)

Since, they are given to be orthogonal circles, we have

2gg

′

+2ff

′

=c+c

′

2(−λ)(

3

−4

)+2(−1)(

6

29

)=−7

⇒

3

8λ

−

3

29

=−7

⇒8λ−29=−21

⇒8λ=8

⇒ λ=1