Question

If the tangents $PQ$ 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 $△PQR$.

• 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

Answer: (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 the point $P(x_1,y_1)$ and hence equation of $QR$ is ${xx}_1+{yy}_1-a^2=0$ ...(2)

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

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

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

i.e., 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

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