Question

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

Answer 1

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

Answer 2

Handwriting is bad because ....Hope u will like