CAT 2023 Slot 1 QA Question 4: If x and y are real numbers such that $x^2 + (x-2y-1)^2 = -4y(x+y)$, then the value of $x - 2y$ is

Question

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

Accepted Answer

Expand and bring everything to the left side.

$x^2 + (x-2y-1)^2 + 4y(x+y) = 0$
$2x^2 + 8y^2 - 2x + 4y + 1 = 0$

Now complete the squares.
$2(x - \dfrac{1}{2})^2 + 8(y + \dfrac{1}{4})^2 = 0$

A sum of squares can be zero only when each square is zero.
$x = \dfrac{1}{2}$ and $y = \dfrac{-1}{4}$

$x-2

If x and y are real numbers such that x2+(x−2y−1)2=−4y(x+y)x^2 + (x-2y-1)^2 = -4y(x+y)x2+(x−2y−1)2=−4y(x+y), then the value of x−2yx - 2yx−2y is