The equation of the common tangent to the parabolas `y^2= 4ax` and `x^2= 4by` is given by

Explanation

To find the equation of common tangent to two curves, we first consider the standard equation of tangent to one of the curves and then use the condition of tangency for the other curve.

Equation of tangent to `y^(2)=4ax` having slope m is
`y=mx+(a)/(m)` (1)
Solving this line with the parabola `x^(2)=4by`, we have
`x^(2)=4b(mx+(a)/(m))`
`orx^(2)-4bmx-(4ab)/(m)=0`
If line (1) touches the parabola `x^(2)=4ay` then above equation has equal roots.
Thus, discriminant is zero.
`or16b^(2)m^(2)+16(ab)/(m)=0`
`rArr" "m^(3)=-(a)/(b)`
`rArr" "m=-((a)/(b))^((1)/(3))`
Thus, equation of common tangent is `y=-((a)/(b))^((1)/(3))x-((b)/(a))^((1)/(3))a`.