Question

The coordinates of two points $P$ and $Q$ are $(2,3)$ and $(3,2)$ respectively. Circles are described on $OP$ and $OQ$ as diameters; $O$ being the origin, then length of the common chord is

• A. $\sqrt{2}$
• B. $\sqrt{13}$
• C. $\frac{5}{\sqrt{2}}$
• D. None of these

Answer 1

The correct option is C $\frac{5}{\sqrt{2}}$

Equation of circles are $x^2+y^2-2x-3y=0$ and $x^2+y^2-3x-2y=0$.
Equation of the common chord is $y=x$, for the points os intersection
${2x}^2-5x=0\Rightarrow x=0,\frac{5}{2}$
Length of the chord $=\sqrt{{\left(\frac{5}{2}\right)}^2+{\left(\frac{5}{2}\right)}^2}=\frac{5}{\sqrt{2}}$