Question

Find the locus of the middle points of all tangents drawn from points on the directrix to the parabola y2=4ax

Answer 1

Answer:

Locus is:     y² (2x + a) = a (3x + a)²

Explanantion:

Let the parabola we consider and draw tangents  be standard parabola

y² = 4ax.

the equation of a parabola whose co-ordinate of the vertex is at (0, 0), the co-ordinates of the focus are (a, 0), the equation of directrix is x = -a or x + a = 0

The locus of that point is     y² (2x + a) = a (3x + a)²

That's the final answer.

I hope it will help you.

Answer 2

Let P = (at²₁ ,2at₁) and Q = (at²₂,2at₂) be the end points of a chord.

Mid point M = (a (t²₁+t²₂)2,a(t₁+t₂))

slope of OP=2/t₁ and slope of OQ=2/t₂

Since OP⊥OQ , 2/t₁ 2/t₂=−1 or t₁t₂=−4

ForM,x=a(t²₁+t²₂)/2=a(t₁+t₂)²–2at₁t₂/2=a(t₁+t₂)²+8a/2

y=a(t₁+t₂)

Eliminating t₁+t₂ we get

y² = 2a(x-4a)