The abscissae of the two points $A$ and $B$ are the roots of the equation $x^{2}+2 a x-b^{2}=0$ and

Question

The abscissae of the two points $A$ and $B$ are the roots of the equation $x^{2}+2 a x-b^{2}=0$ and their ordinates are the roots of the equation $x^{2}+2 p y-q^{2}=0$. Find the equation of the circle on $A B$ as diameter.

Professor PC Thomas Classes · Answer

Let $A=\left(x_{1}, y_{1}ight), B=\left(x_{2}, y_{2}ight)$ Then $x_{1}+x_{2}=-2 a ; y_{1}+y_{2}=-2 p$ $$ x_{1} x_{2}=-b^{2} ; \quad y_{1} y_{2}=-q^{2} $$ The equation of the circle on $A B$ as diameter is, $\left(x-x_{1}ight)\left(x-x_{2}ight)+\left(y-y_{1}ight)\left(y-y_{2}ight)=0$\ or $\quad x^{2}+y^{2}-x\left(x_{1}+x_{2}ight)-y\left(y_{1}+y_{2}ight)+x_{1} x_{2}+$ $y_{1} y_{2}=0$\ $	herefore \quad x^{2}+y^{2}-x(-2 a)-y(-2 p)-\left(b^{2}+q^{2}ight)=0$ or $\quad x^{2}+y^{2}+2 a x+2 p y-\left(b^{2}+q^{2}ight)=0$\