Question : If $x^2+y^2+2 y+4 x+5=0$, then $\frac{x+y}{x-y}=$______.
Option 1: $-3$
Option 2: $\frac{1}{3}$
Option 3: $-1$
Option 4: $3$
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Correct Answer: $3$
Solution :
Given equation, $x^2+y^2+2 y+4 x+5=0$
Let us try to make it a perfect square.
$x^2+4 x+y^2+2 y+5=0$
⇒ $x^2 + 4x + 4 -4 + y^2 +2y + 1 -1 + 5=0$
⇒ $(x+2)^2 + (y+1)^2 -5 + 5 = 0$
⇒ $(x+2)^2 + (y+1)^2 = 0$
⇒ $(x+2)^2=0$ and $(y+1)^2 = 0$ [If the sum of two squares is zero then each term individually will also be zero]
⇒ $x+2 = 0$ and $y+1 = 0$
⇒ $x=-2$ and $y=-1$
$\therefore$ $\frac{x+y}{x-y}=\frac{-2-1}{-2+1}=\frac{-3}{-1}=3$
Hence, the correct answer is 3.
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