Question

A circle passing through $(0,0),(2,6),(6,2)$ cuts the x axis a the point $P\ne (0,0)$. Then the length of OP, where O is origin, is

• A. $\frac{5}{2}$
• B. $\frac{5}{\sqrt{2}}$
• C. $5$
• D. $10$

Answer 1

The correct option is C $5$

Let $x^2+y^2+2gx+2fy=0$ be the equation of circle passing through origin

Given that it also passes through $(2,6)$ and $(6,2)$

$⟹4+36+4g+12f=0$

$⟹g+3f+10=0–eq.1$

And

$⟹36+4+4f+12fg=0$

$⟹f+3g+10=0–eq.2$

Solving eq.1 and eq.2

$f–g=\frac{-5}{2}$

Equation is $2x^2+2y^2–10x-10y=0$

Substituting $y=0$

$2x^2–10x=0⟹x=5,0$

$P=(5,0),O=(0,0)$

$OP=5$