If x² + y² + 2y + 4x + 5 = 0, then \(\frac{x+y}{x-y}=\) ________.

Given:

x2 + y2 + 2y + 4x + 5 = 0

Concept Used:

The concept of quadratic equation is used

Calculations:

The given equation is 

x2 + y2 + 2y + 4x + 5 = 0

To make the equation quadratic for x and y we have to add some values according to the coefficient of x  =and y respectively

⇒ The coefficient of x is 4x = 2 × 2 × x 

so for x, we have to add square of 2 and subtract square of 2

⇒ The coefficient of y is 2x = 2 × 1 × y 

so for x, we have to add square of 1 and subtract square of 1

So, The equation will become

⇒ x2 + y2 + 2y + 4x + 5 + 4 - 4 + 1 - 1 = 0

⇒ x2 + 4x + 4 + y2 + 2y + 1 + 5 - 4 - 1= 0

⇒ (x + 2)² + (y + 1)² = 0

On putting the value of x = -2 and y = -1, we get the value 0

⇒ So , x = - 2 and y = -1

The value of \(\frac{x+y}{x-y}=\)

⇒ \(\frac{x+y}{x-y}=\frac{-2-1}{-2 - (-1)}=\frac{-3}{-1} \) = 3

⇒ Hence, The value of the given equation is 3

Alternate Method

x2 + y2 + 2y + 4x + 5 = 0

⇒ x2 + y2 + 2y + 4x + 4 + 1 = 0 

⇒ x2 +  4x + 4 + y2 + 2y + 1 = 0 

⇒ (x + 2)2 + (y + 1)2 = 0

So, x = -2 and y = -1

Then, \(\frac{x+y}{x-y}=\) -3/-1 = 3