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### AIEEE 2002

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
$$y = \pm \left( {x + a} \right)$$

## Explanation

Any tangent to the parabola $${y^2} = 8ax$$ is

$$y = mx + {{2a} \over m}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)$$

If $$(i)$$ is a tangent to the circle, $${x^2} + {y^2} = 2{a^2}$$ then,

$$\sqrt {2a} = \pm {{2a} \over {m\sqrt {{m^2} + 1} }}$$

$$\Rightarrow {m^2}\left( {1 + {m^2}} \right) = 2$$

$$\Rightarrow \left( {{m^2} + 2} \right)\left( {{m^2} - 1} \right) = 0$$

$$\Rightarrow m = \pm 1.$$

So from $$(i),$$ $$y = \pm \left( {x + 2a} \right).$$

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