Question

Tangents are drawn to the circle $x^2+y^2=1$ at the points where it is met by the circles, $x^2+y^2-(\lambda +6)x+(8-2\lambda )y-3=0,\lambda$ being the variable. The locus of the point of intersection of these tangents is

• A. $2x-y+10=0$
• B. $x+2y-10=0$
• C. $x-2y+10=0$
• D. $2x+y-10=0$

Answer 1

The correct option is A $2x-y+10=0$

Locus of point of intersection of tangents chord of contact of $(x_1,y_1)$ w.r.t.

$x^2+y^2=1$ is $x{x}_1+y{y}_1=1\left(AB\right)$ ........... (i)

AB is also a common chord between two circles

$\therefore -1+(\lambda +6)x-(8-2\lambda )y+3=0$

$\Rightarrow (\lambda +6)x-(8-2\lambda )y+2=0$ ....... (ii)

Comparing Equations (i) and (ii), we get

$\frac{x_1}{\lambda +6}=\frac{y_1}{2\lambda -8}=\frac{-1}{2}$

Eliminate $\lambda \Rightarrow 2x-y+10=0$ which is required locus.