Find the equation of the common tangent of parabola y^{2}=4ax and x^{2}=4by.

Let the equation of common tangent to *y*^{2} = 4*ax* and *x*^{2} = 4 by be *y* = *mx *+ *c*.

*y* = *mx *+ *c* is the tangent to the parabola *y*^{2} = 4*ax.*

* *is tangent to the parabola *x*^{2} = 4 by, then it will cut the parabola *x*^{2} = 4 by in two coincidental points.

*mx*^{2} = 4*bm*^{2}*x* + 4*ab*

*mx*^{2} – 4*bm*^{2}*x* – 4*ab *= 0

∴ D = (– 4*bm*^{2})^{2} – 4 × *m* × (– 4*ab*) = 0

Equation of the common tangent is

Thus, the equation of the common tangent is .

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