Question

Answer the following:

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(`"at"_1^2`, 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(`"at"_1^2`, 2at₁) is 

yt₁ = x + `"at"_1^2`   ...(i)

∵ Point Q lie on tangent

∴ put x = 0 in equation (i)

yt₁ = `"at"_1^2`

y = at₁

∴ Co-ordinate of point Q(0, at₁)

S = (a, 0), P(`"at"_1^2`, 2at₁), Q(0, at₁)

Slope of SQ = `(y_2 - y_1)/(x_2 - x_1)`

= `("at"_1 - 0)/(0 - "a")`

= `("at"_1)/(-"a")`

= – t₁

Slope of PQ = `(y_2 - y_1)/(x_2 - x_1)`

= `(2"at"_1 - "at"_1)/("at"_1^2)`

= `"at"_1/"at"_1^2`

= `1/"t"_1`

∵ Slope of SQ × Slope of PQ

= `-"t"_1 xx 1/"t"_1`

= – 1

∴ SP subtends a right angle at Q.