The equation of tangent of a parabola

Question

Equation of tangent at point P(x₁, y₁) to parabola y₂ = 4ax.

Answer 1

Equation of tangent at point P(x₁, y₁) to parabola y² = 4ax

yy₁ – 2a(x + x₁) = 0 or T = 0 ...(5) 

where T is an expression which we get by replacing y² by yy₁ and 2x by x + x₁. 

Equation of tangent at point P(t) or P(at², 2at) In (5) replacing y₁ by 2at and x₁ by at², we have equation of tangent, 

2at y = 2a(x + at²) or ty = x + at² ....(6) 

Here slope of tangent m = 1/t. 

Equation of tangent in slope (m) form In (6) replacing t by 1/m we have y = mx + a/m which is equation of tangent in terms of slope and the point of contact is (a/m², 2a/m). 

Thus if line y = mx + c touches parabola y² = 4ax we must have c = a/m (comparing equation with y = mx + a/m).