Question

The tangent at point P on the parabola y² = 4ax meets the Y-axis in Q. If S is the focus, show that SP subtends a right angle at Q.

Answer 1

Let P(at₁² , 2at₁) be a point on the parabola and S(a, 0) be the focus of parabola y² = 4ax

Since the tangent passing through point P meet Y-axis at point Q, equation of tangent at P (at²₁, 2 at₁) is 

yt₁ = x + at²₁,…..(i)

∴ Point Q lie on tangent 

∴ put x = 0 in equation (i)

yt₁ = at²₁

y = at₁

∴ Co-ordinate of point Q(0, at₁)

S = (a, 0), P(at²_{1, 2}at₁)

∴ SP subtends a right angle at Q.