Find the coordinates of the center and foci and the lengths of the major and minor axes for the ellipse of the

1. 36x^2 + 81y^2 = 2916

2. 16x^2 + 25y^2 + 32x - 150y = 159

1.

36x² + 81y² = 2916
36x²/2916 + 81y²/2916 = 1
x²/81 + y²/36 = 1

Centre: (0, 0)

a = 9 ----> Length of major axis = 2a = 18
b = 6 ----> Length of minor axis = 2b = 12
c = √(81−36) = √45 = 3√5

Major axis is along x-axis, so foci = (0±c, 0) = (±3√5, 0)

2.

16x² + 25y² + 32x − 150y = 159
16 (x² + 2x) + 25 (y² − 6y) = 159
16 (x² + 2x + 1) + 25 (y² − 6y + 9) = 159 + 16(1) + 25(9)
16 (x + 1)² + 25 (y − 3)² = 400
(x + 1)²/25 + (y − 3)²/16 = 400

Centre: (−1, 3)

a = 5 ----> Length of major axis = 2a = 10
b = 4 ----> Length of minor axis = 2b = 8
c = √(25−16) = √9 = 3

Major axis is parallel to x-axis, so foci = (−1±c, 3) = (−4, 3) and (2, 3)