$\frac{x}{a} + \frac{y}{b} = 1 $ and $\frac{x}{c} + \frac{y}{d} = 1 $ intersect the axes at four concyclic points…

Problem: $\large \frac{x}{a} + \frac{y}{b} = 1 $ and $\large \frac{x}{c} + \frac{y}{d} = 1 $ intersect the axes at four concyclic points and c2 + a2 = b2 + d2, then these lines can intersect at (a, b, c, d > 0)

(A) (1, 1)

(B) (1, – 1)

(C) (2, – 2)

(D) (3, – 2)

Ans: (A) , (B), (C)

Sol: Points A, B, C, D are concyclic, then ac = bd.

mcq math

The co-ordinate of the points of intersection of lines are

$\large (\frac{ac(b-d)}{bc-ad}) , (\frac{bd(c-a)}{bc-ad}) $

given c2 + a2 = b2 + d2    ( ac = bd)

⇒ (c – a)2 = (b – d)2

⇒ c – a = ± (b – d)

then the locus of the points of intersection is y = ± x.