Question

If the abscissae and ordinates of two points P and Q are roots of the equations x² + 2ax − b² = 0 and x² + 2px − q² = 0 respectively, then write the equation of the circle with PQ as diameter.

Answer 1

The roots of the equations x² + 2ax − b² = 0 and x² + 2px − q² = 0 are \[- a \pm \sqrt{a^2 + b^2}\]  and \[- p \pm \sqrt{p^2 + q^2}\] . Therefore, the coordinates of P and Q are \[\left( - a + \sqrt{a^2 + b^2}, - p + \sqrt{p^2 + q^2} \right) \text { and } \left( - a - \sqrt{a^2 + b^2}, - p - \sqrt{p^2 + q^2} \right)\]   -a+a2+b2, -p+p2+q2 and -a-a2+b2, -p-p2+q2 , respectively.
So, the required equation of the circle is

\[\left( x + a - \sqrt{a^2 + b^2} \right)\left( x + a + \sqrt{a^2 + b^2} \right) + \left( y + p - \sqrt{p^2 + q^2} \right)\left( y + p + \sqrt{p^2 + q^2} \right) = 0\]

\[\Rightarrow \left( x + a \right)^2 - a^2 - b^2 + \left( y + p \right)^2 - p^2 - q^2 = 0\]

\[x^2 + y^2 + 2ax + 2yp - p^2 - q^2 = 0\]