A) \[\left( \frac{p}{2},p \right)\]
B) \[(2p,2p)\]
C) \[\left( -\frac{p}{2},p \right)\]
D) \[\left( -\frac{p}{8},\frac{p}{2} \right)\]
Correct Answer: A
Solution :
| The focus of the parabola is at \[\left( \frac{p}{2},0 \right)\] and directrix is \[x=-\frac{p}{2}\] |
| Centre of the circle is \[\left( \frac{p}{2},0 \right)\] and radius \[=\frac{p}{2}-\left( -\frac{p}{2} \right)=p\] |
| Equation of circle is \[{{\left( x-\frac{p}{2} \right)}^{2}}+{{\left( y-0 \right)}^{2}}={{p}^{2}}\]\[\Rightarrow \]\[{{x}^{2}}+{{y}^{2}}-px-\frac{3{{p}^{2}}}{4}=0\] |
| solving this equation with \[{{y}^{2}}=2px,\] we obtain \[x=\frac{p}{2}\]and\[y=\pm p\] |
| \[\therefore \]the points of intersection of the circle and the parabola are \[\left( \frac{p}{2},p \right)\] and \[\left( \frac{p}{2},-p \right).\] |
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