21 find the coordinates of the center of the ellipse represe

21. find the coordinates of the center of the ellipse represented by 4x^2+9y^2-18y-27=0

Solution: Ellipse is 4x^2+9y^2-18y-27=0.

[ Completing the square and write in the form of

(x-h)^2/a^2+(y-k)^2/b^2 =1, then find the coordinates of the center .]

Thus 4x^2 + 9y^2 - 18y - 27 = 0

or 4x^2 + 9y^2 - 18y + 3^2 - 3^2 - 27 = 0

or 4x^2 + (3y^2 - 3)^2 - 9 - 27 = 0

or 4x^2 + ( (3y)^2 - 3)^2 - 36 = 0

or 4x^2 + 3^2(y^2 - 1)^2 = 36

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

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

Comparing with (x-h)^2/a^2+(y-k)^2/b^2 =1, we get

h = 0, and k = 1.

centre (h, k) = (0, 1)

Therefore, coordinates of the center is (0, 1). Ans