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 $\left(-a+\sqrt{a^2+b^2},-p+\sqrt{p^2+q^2}\right)\mathrm{and}\left(-\mathrm{a}-\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2},-\mathrm{p}-\sqrt{{\mathrm{p}}^2+{\mathrm{q}}^2}\right)$, respectively.

So, the required equation of the circle is $\left(x+a-\sqrt{a^2+b^2}\right)\left(x+a+\sqrt{a^2+b^2}\right)+\left(y+p-\sqrt{p^2+q^2}\right)\left(y+p+\sqrt{p^2+q^2}\right)=0$.
$\Rightarrow {\left(x+a\right)}^2-a^2-b^2+{\left(y+p\right)}^2-p^2-q^2=0$
$x^2+y^2+2ax+2yp-p^2-q^2=0$