ELLIPSE QUESTION! PLEASE HELP!?

Question

Find the center, vertices, foci, and eccentricity of the ellipse.
25x^2 + 4y^2 + 100x − 40y + 100 = 0

Answer 1

The ellipse equation 25x² + 4y² + 100x - 40y + 100 = 0

25x² + 100x + 4y² - 40y = - 100

25(x² + 4x) + 4(y² - 10y) = - 100

To change the expressions (x² + 4x) and (y² - 10y) into a perfect square trinomial,

add (half the x coefficient)² and add (half the y coefficient)²to each side of the equation.

(half the x coefficient)² = (4/2)² = 4

(half the y coefficient)² = (-10/2)² = 25

25(x² + 4x + 4) + 4(y² - 10y + 25) = - 100 + 100 + 100

25(x + 2)² + 4[y - (1/2)]² = 100

[25(x + 2)²]/100 + [4(y - 5)²]/100 = 100

[x - (-2)]²/2² + [y - 5]²/5² = 1

Compare the standard form of ellipse

a² > b²

If the larger denominator is under the "y" term, then the ellipse is vertical.Center (h, k ).

a = length of semi-major axis, b = length of semi-minor axis.

Center (h, k) = ( - 2, 5)

a  = 5, b  = 2.

Vertices are (h, k + a) , (h, k - a)

Vertices (-2, 0) and (-2, 10)

Foci (h, k + c)(h, k - c)

c = √(a² - b²)

c = √(25 - 4)

c = √21

Foci (- 2, 5 + √21), (- 2, 5 - √21)

Eccentricity e = c/a

e = √21/5.