cat 2023 Question 2

Question

If x and y are real numbers such that $x^2+(x-2 y-1)^2=-4 y(x+y)$, then the value x−2y is

Accepted Answer

Given,

$ x^{2} + (x - 2y - 1)^{2} = -4y(x + y) $

$ x^2 + 4xy + 4y^2 + (x-2y-1)^2 = 0 $

$ (x+2y)^2 + (x-2y-1)^2 = 0 $

For the L.H.S. of the equation to be 0, each of the square terms should be 0 (as squares cannot be negative)

$ x - 2y - 1 = 0 $ => $ x - 2y = 1 $

Question 2.