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).
