find the equations of common tangent to the parabola y²⁼4x and x²=4y.

Hi,

Tangent to first parabola is y=mx+1/m
If this is also the tangent to second parabola than put this equation in second parabola.
Form a quadratic in x and put its d=0 as tangent and curve has only one common solution that is the point of tangency.

Get the values of m. Put all thoes values of m in the tangent, you will get the common tangents.