Question

If the abscissae and ordinates of two points P and Q are roots of the equations $x^2+2ax-b^2=0$
and $x^2+2px-q^2=0$ respectively,
then write the equation of the circle with PQ as diameter.

Answer 1

Let $\alpha ,\beta$ and r, $\delta$ are the roots of the first and second given equation,
so
$\alpha +\beta =—2a,\alpha \beta =-b^2\gamma +\delta =-2p,\gamma \delta =-q^2$
Coordinates of P and Q are $(\alpha ,\gamma )$ and $(\beta ,\delta )$ respectively
The equation of circle on PQ as diameter is
$(x-\alpha )(x-\beta )+(y-\gamma )(y-\delta )=0x^2+y^2-(\alpha +\beta )x-(\gamma +\delta )y+\alpha \beta +\gamma \delta =0x^2+y^2+2ax+2py-b^2-q^2=0$