• # question_answer Consider a circle with its 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                           [IIT 1995] A)            $\left( \frac{p}{2},\ p \right)$ B)            $\left( \frac{p}{2},\ -p \right)$ C)            $\left( \frac{-p}{2},\ p \right)$       D)            $\left( \frac{-p}{2},\ -p \right)$

Focus of parabola ${{y}^{2}}=2px$ is                   $(p/2,0)$                                                         .....(i)                   \ Radius of circle whose centre is $(p/2,0)$and touching $x+(p/2)=0$is p.                   Equation of circle is ${{\left( x-\frac{p}{2} \right)}^{2}}+{{y}^{2}}={{p}^{2}}$     .....(ii)                    From (i) and (ii), we get the point of intersection $\left( \frac{p}{2},p \right),\left( \frac{p}{2},-p \right)$.