Question

A circle with centre lying on the focus of the parabola $y^2=2px$ such that it touches the directrix of the parabola. Then a point of intersection of the circle and the parabola is

• A. $(\frac{p}{2},p)$
• B. $(\frac{p}{2},2p)$
• C. $(-\frac{p}{2},p)$
• D. $(-\frac{p}{2},-p)$

Answer 1

The correct option is A $(\frac{p}{2},p)$

A circle with focus of parabola $y^2=2px$ and touches directrix.

Then point of intersection of circle and parabola is:

Circle has radius $\sqrt{{(\frac{p}{2}+\frac{p}{2})}^2+0}=p$

Focus $(\frac{p}{2},0)$

Directrix $x=\frac{-p}{2}$

Equation of circle is

${(x-\frac{p}{2})}^2+y^2=p^2{x}^2+y^2-px=\frac{3p^2}{4}x=\frac{p}{2}y=p$

Point $=(\frac{p}{2},p)$