Question

If the tangent $PQ$ and $PR$ are drawn to the circle $x^2+y^2=a^2$ from the point $P(x_1,y_1)$, then the equation of the circumcircle of $△PQR$ is

• A. $x^2+y^2-x{x}_1-y{y}_1=0$
• B. $x^2+y^2+x{x}_1+y{y}_1=0$
• C. $x^2+y^2-2x{x}_1-2y{y}_1=0$
• D. None of these

Answer 1

The correct option is A $x^2+y^2-x{x}_1-y{y}_1=0$

Given circle is $x^2+y^2-a^2=0$ ...(1)

Since $PQ$ and $PR$ are tangents to the circle (1), therefore $QR$ is chord of contact of point $P(x_1,y_1)$ and hence equation of $QR$ is $x{x}_1+y{y}_1-a^2=0$ ...(2)

Now, equation of any circle through the point of intersection $Q$ and $R$ of circle (1) and line (2) is

$x^2+y^2-a^2+k(x{x}_1+y{y}_1-a^2)=0$ ...(3)

Circle (3) will be circumciecle of $△PQR$ it it passes through the point $P(x_1,y_1)$ if

${x_1}^2+{y_1}^2-a^2+k({x_1}^2+{y_1}^2-a^2)=0\Rightarrow k=-1$

Hence from (3), equation of required circle is

$x^2+y^2-a^2-1(x{x}_1+y{y}_1-a^2)=0{\Rightarrow x}^2+y^2-x{x}_1-y{y}_1=0$