ellipses conic sections

Question

find the center and foci of the ellipse

4x²+9y²+8x-36y+4=0

Answer 1

4x² + 9y² + 8x - 36y + 4 = 0

First, get this into the standard form: (x-j)² / a² + (y-k)² / b² = 1

Start by completing the square for the x and y terms:

4x² + 8x + 4 =

4* (x² + 2x) = 4* (x + 1)²

9y² - 36y =

9* (y² - 4y + 4) = 9* (y - 2)² +36

Combine and move constant to right side:

4* (x + 1)² + 9* (y - 2)² = 36

Divide both sides by 36:

(x + 1)² /9 + (y - 2)² /4 = 1

So: a = 3 and b = 2

The center is at (-1, 2)

The distance to the focus is f = √(a2 - b2) = √(9 - 4) = √5

Answer 2

We grouping and compleet squares: (4x² + 8x + 4) + (9y² - 36y + 36) = -4 + 4 + 36,

4(x + 1)²+ 9(y - 2)² = 36 or (x + 1)²/9 + (y - 2)²/4 = 1.

(-1,2) is center of ellipse,

c² = a² - b² = 9 - 4 = 5, c = 5^1/2,

Foci: (±5^1/2 - 1, 2)