Question

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

• A. $x^2+y^2+2ay+2px-b^2-q^2=0$
• B. $x^2+y^2+2ax+2py-b^2-q^2=0$
• C. $x^2+y^2+2(a+p)x-b^2-q^2=0$
• D. $x^2+y^2+2(a+p)y-b^2-q^2=0$

Answer 1

The correct option is B $x^2+y^2+2ax+2py-b^2-q^2=0$

Let $P(x_1,y_1),Q(x_2,y_2)$ be end points of diameter.
Given
$x_1,x_2$ are the roots of $x^2+2ax-b^2=0$
$y_1,y_2$ are the roots of $x^2+2px-q^2=0$
$\Rightarrow x_1+x_2=-2a,x_1{x}_2=-b^2$
$\Rightarrow y_1+y_2=-2p,y_1{y}_2=-q^2$
Equation of circle having $PQ$ as diameter is
$(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0$
$\Rightarrow x^2-(x_1+x_2)x+x_1{x}_2+y^2-(y_1+y_2)y+y_1{y}_2=0$
$\Rightarrow x^2+y^2+2ax+2py-b^2-q^2=0$