Question
Two common tangents to the circle $${x^2} + {y^2} = 2{a^2}$$ and parabola $${y^2} = 8ax$$ are-
A.
$$x = \pm \left( {y + 2a} \right)$$
B.
$$y = \pm \left( {x + 2a} \right)$$
C.
$$x = \pm \left( {y + a} \right)$$
D.
$$y = \pm \left( {x + a} \right)$$
Answer :
$$y = \pm \left( {x + 2a} \right)$$
Solution :
Any tangent to the parabola $${y^2} = 8ax$$ is
$$y = mx + \frac{{2a}}{m}.....({\text{i}})$$
If (i) is a tangent to the circle, $${x^2} + {y^2} = 2{a^2}$$ then,
$$\eqalign{
& \sqrt 2 a = \pm \frac{{2a}}{{m\sqrt {{m^2} + 1} }} \cr
& \Rightarrow {m^2}\left( {1 + {m^2}} \right) = 2 \cr
& \Rightarrow \left( {{m^2} + 2} \right)\left( {{m^2} - 1} \right) = 0 \cr
& \Rightarrow m = \pm 1 \cr} $$
So from (i), $$y = \pm \left( {x + 2a} \right)$$