Question

If the normals at two points p and q of a parabola y2=4ax intersect at a third point r on the curve then find the product of ordinates of p and q.

Answer 1

Answer:

8a²

Step-by-step explanation:

Let the points p be t₁ and q be t₂ i.e., (at₁², 2at₁) and (at₂², 2at₂).

We know that equaation of normat at pt(t) is given by tx+y = at³+2at

Let the point of intersection of 2 normals at Pt(t₁) and Pt(t₂) be Pt(t₃).

Now, also if the normal at any point(t₁) intersects at any  point(t) on a parabola, then t = -t₁-2/t₁----(*)

Using (*),normal at  point(t₁) intersects at  point(t₃) on a parabola,

so t₃ = -t₁-2/t₁------(1)

Using (*),normal at  point(t₂) intersects at  point(t₃) on a parabola,

so t₃ = -t₂-2/t₂-----(2),

By comparing (1) and (2) we get,

-t₁-2/t₁ = -t₂-2/t₂

=>(t₁ -t₂) = 2/t₂-2/t₁

=>t₁t₂ = 2.

Product of ordinated of P and Q is (2at₁)(2at₂) = 4a²t₁t₂=8a².