Question

2x^2 + 2y^2 - 8x + 10y + 2 =0

Answer 1

A circle of radius r centered at (a, b) has equation

(x - a)² + (y - b)² = r ²

To get the given equation in this form, complete the square with respect to both x and y.

2x² - 8x = 2 (x² - 4x)

2x² - 8x = 2 (x² - 4x + 4 - 4)

2x² - 8x = 2 ((x - 2)² - 4)

2x² - 8x = 2 (x - 2)² - 8

2y² + 10y = 2 (y² + 5y)

2y² + 10y = 2 (y² + 5y + 25/4 - 25/4)

2y² + 10y = 2 ((y + 5/2)² - 25/4)

2y² + 10y = 2 (y + 5/2)² - 25/2

So we have

2x² + 2y² - 8x + 10y + 2 = 0

2 (x - 2)² - 8 + 2 (y + 5/2)² - 25/2 + 2 = 0

2 (x - 2)² + 2 (y + 5/2)² = 37/2

(x - 2)² + (y + 5/2)² = 37/4

(x - 2)² + (y + 5/2)² = ((√37)/2)²

which means this circle has radius (√37)/2 and center (2, -5/2).