The line y=mx is tangent to the curve y^2 = x-c. show that c = 1/ (4m^2)

Hello,
Let (a,b) the intersection of the tangent and the curve.
That point must be double.

So
 b=m*a
b²=a-c
==>(ma)²-a+c=0
==>m²a²-a+c=0 roots must be equal
Δ=(-1)²-4*m²c=0 => 4m²c=1==>c=1/(4m²)