Question

If the chord of contact of the tangents from the point $\left(\alpha , \beta \right)$ to the circle $x^{2}+y^{2}=r_{1}^{2}$ is a tangent to the circle $\left(x - a\right)^{2}+\left(y - b\right)^{2}=r_{2}^{2}$ , then

• A. $r_{2}^{2}\left(\alpha^{2}+\beta^{2}\right)=\left(r_{1}^{2}-a \alpha-b \beta\right)^{2}$
• B. $r_{2}^{2}\left(\alpha^{2}+\beta^{2}\right)=\left(r_{1}^{2}+a \alpha-b \beta\right)^{2}$
• C. $r_{2}^{2}\left(\alpha^{2}+\beta^{2}\right)=\left(r_{1}^{2}-a \alpha+b \beta\right)^{2}$
• D. $r_{2}^{2}\left(\alpha^{2}+\beta^{2}\right)=\left(r_{1}^{2}+a \alpha+b \beta\right)^{2}$

Answer 1

Answer: (A)

The equation of the chord of contact of tangents drawn from a point $\left(\alpha , \beta \right)$ to the circle $x^{2}+y^{2}=r_{1}^{2}$ is
$\alpha x+\beta y=r_{1}^{2}$
This line touches the circle $\left(x - a\right)^{2}+\left(y - b\right)^{2}=r_{2}^{2}$
$\therefore \left|\frac{a \alpha + b \beta - r_{1}^{2}}{\sqrt{\alpha ^{2} + \beta ^{2}}}\right|=r_{2}$
$\Rightarrow r_{2}^{2}\left(\alpha^{2}+\beta^{2}\right)=\left(a \alpha+b \beta-r_{1}^{2}\right)^{2}$
$\Rightarrow r_{2}^{2}\left(\alpha^{2}+\beta^{2}\right)=\left(r_{1}^{2}-a \alpha-b \beta\right)^{2}$