Find the locus of middle points of a family of focal chords of the parabola y^2=4ax Class: 11. Created by Explode. Math

Find the locus of middle points of a family of focal chords of the parabola y^2=...

rohitkumargupta HELLO DEAR ,
Let the equation of the parabola be y2 = 4ax.

Let t1, t2 be the extremities of the focal chord. Then t1 . t2 = – 1.

The equation of the circle on t1, t2 as diameter is

(x – at22) (x – at22) + (y – 2at1) (y – 2at2) = 0

or x2 + y2 – ax (t12 + t22) – 2ay (t1 + t2) + a2 t12t12 + 4a2 t1t2 = 0

⇒ x2 + y2 – ax (t12 + t22) – 2ay (t1 + t2) – 3a2 = 0. (? t1t2 = –1)

If (α,β) be the centre of the circle, then α = a/2 (t12+t22 ) If (α, β) be the centre of the circle, then α = a/2 (t12+t22)

β = a (t1 + t2) ⇒ (t1 + t2)2 = β2/a2 ⇒ t12 + t22 + 2t1t2 =β2/a2 ⇒ 2α/a-2= β2/a2

⇒ 2aα – 2a2 = β2 ⇒ β2 = 2a (α – a).

Hence locus of (α, β) is y2 = 2a(x – a).
hope u understand

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Explode (a , 2a) and (a ,-2a)

Explode this points are satisfied with the equation of (y-y1)/(x-x1) = (y1-y2)/(x1-x2)

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