Question

Find the equation of common tangent(s) of (i) Parabola $y^2 = 4ax$ and $x^2 = 4by$.

Answer 1

The equation of the common tangent is $(a/b)^{1/3} x + y + a(b/a)^{1/3} = 0$.

Answer 2

Let the common tangent be $Ax+By+C=0$. The condition for this line to be tangent to $y^2=4ax$ is $aB^2=AC$. The condition for this line to be tangent to $x^2=4by$ is $bA^2=BC$. From the first condition, $C = aB^2/A$. Substituting this into the second condition, $bA^2 = B(aB^2/A) \implies bA^3 = aB^3 \implies (A/B)^3 = a/b$. Let $k = A/B$, so $k^3 = a/b$. From $C=aB^2/A$, dividing by $B$, $C/B = a(B/A) = a/k$. The equation of the tangent $Ax+By+C=0$, when divided by $B$, becomes $(A/B)x + y + (C/B) = 0$, which is $kx + y + a/k = 0$. Substituting $k=(a/b)^{1/3}$, we get $(a/b)^{1/3}x + y + a(b/a)^{1/3} = 0$. This is the equation of the common tangent.