Find the relation between m and c if the line y = mx + c is a tangent to the curve y^2 = 2x?

The line: y = mx + c ...... [1]
The curve: y² = 2x ...... [2]

Put [1] into [2]:
(mx + c)² = 2x
m²x² + 2mcx + c² = 2x
m²x² + (2mc ‒ 2)x + c² = 0

As the line is a tangent to the curve, there is only one point of contact.
The above equation should have only one real root, and thus :
The discriminant, Δ = 0
(2mc ‒ 2)² ‒ 4 × m² × c² = 0
4m²c² ‒ 8mc + 4 ‒ 4m²c² = 0
8mc = 4
2mc = 1
m = 1/(2c)

[ mx + c ]² = 2x
m²x² + 2mcx + c² = 2x
m²x² + [ 2mc - 2 ] x + c² = 0

For tangency
[ 2mc - 2 ]² = 4m² c²
4m²c² - 8mc + 4 = 4m² c²
4 = 8mc
mc = 1/2

y^2 = 2x
differentiate both sides with respect to x
2y dy/dx = 2
dy/dx = 1/ (2y)

slope = 1/(2y)
m=1/(2y)
2y =1/m
y= 1/(2m)

y=mx+c
y-c = m(x-0)
Equation of the tangent at (0,c)

1/ (2m) = mx + c
1/(2m) = m(0) + c

c=1/ (2m)