Two parabolas y^2 = 4a(x – I1 ) and x^2 = 4a(y– I2 ) always touch one another, the quantities l1 and l2 are both variable.

Question

Two parabolas y² = 4a(x – I₁ ) and x² = 4a(y– I₂ ) always touch one another, the quantities I₁ and I₂ are both variable. Locus of their point of contact has the equation 

(A) xy=a² 

(B) xy = 2a² 

(C) xy = 4a² 

(D) none

Answer 1

Correct Answer is: (C) xy = 4a²  

y² = 4a(x = l₁) x² = 4a (y – l₂)

let the POC (h, k)

2yy’ = 4a 2x = 4ay’

(1) and (2) are equal = h/2a ....(2)

a2/k = h/2a

hk = 4a²

xy = 4a²