What is the center, foci, and the lengths of the major and minor axes for the ellipse,

Question

(x+3)^2/36 + (y-4)^2/9=1.

Answer 1

The equation of ellipse is (x + 3)²/36 + (y - 4)²/9 = 1.

The standard form of the equation of an ellipse center (h, k) with major and minor axes of lengths 2a and 2b (where 0 < b < a) is (x - h)²/a² + (y - k)²/b² = 1 or (x - h)²/b² + (y - k)²/a² = 1.

The vertices and foci lie on the major axis, a and c units, respectively, from the center (h, k)  and the relation between a, b and c is c² = a² - b².

Because the denominator of the x² - term (36) is larger than the denominator of the y² - term (9), the major axis is horizontal.

Compare the equation (x + 3)²/36 + (y - 4)²/9 = 1 with (x - h)^2/a^2 + (y - k)^2/b^2 = 1.

a² = 36, b² = 9, h = - 3 and k = 4

a = 6 and b = 3.

To find the value of c, substitute the value of a² = 36 and b² = 9 in c² = a² - b².

c² = 36 - 9

c² = 27

c = ± 3√3.

Center = (h, k ) = (- 3, 4).

Foci = (h ± c, k ) = (- 3 ± 3√3, 4) = (- 3 + 3√3, 4) and (- 3 - 3√3, 4).

The major axes of lengths 2a = 2(6) = 12.

The minor axes of lengths 2b = 2(3) = 6.