The slope of the tangent which touches both the parabolas $y^2 = 4ax$ and the parabola $ x^2=-32y$

Question

The slope of the tangent which touches both the parabolas $y^2$ = $4ax$ and the parabola $x^2=-32y$ how do we find the slope of common tangent if I assume the slope of one of the cords and I find the relation that would hold between the two or should start some other way round because I cannot understand how would this happen thanks in advance .

Answer 1

Hint: A tangent to the parabola $y^2=4ax$ is of the form: $$y=mx+\frac{a}{m}$$ As this must also be the tangent to the second parabola $x^2=-32y$ , the discriminant of the quadratic equation formed when we put $y=mx+\frac{a}{m}$ in $x^2=-32y$ must be $0$.