Find the equations to the common tangents of the circle $x^{2} + y^{2} + 4 a x = 0$ and parabola $y^{2} = 4 a x .$

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Solution

$x^{2} + 4 a x + y^{2} = 0 x^{2} + 4 a x + 4 a^{2} + y^{2} = 4 a^{2} {(x + 2 a)}^{2} + y^{2} = 4 a^{2} . . . . . . (i) y^{2} = 4 a x . . . . . . . . . (i i)$
By tracing the circle and the parabola we can clearly see that they touch each other along the $y$ axis
So the equation of common tangent is $x = 0$ .

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