please solve the problem

Question

the equation of an ellipse is given. find the coordinates of the center and state whether the major axis is vertical or horizontal. x^2/5 + y^2/20 = 1
(x-4)^2/42 + (y+6)^2/23 = 1

Answer 1

The two equations of ellipse are x^2/5 + y^2/20 = 1 and (x - 4)^2/42 + (y + 6)^2/23 = 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/a^2 + (y - k)^2/b^2 = 1 or (x - h)^2/b^2 + (y - k)^2/a^2 = 1.

Compare the equation (x - 0)^2/(√5)^2 + (y - 0)^2/(√20)^2 = 1 with (x - h)^2/b^2 + (y - k)^2/a^2 = 1.

Center = (h, k ) = (0, 0), b = √5 and a = √20.

The major axes of lengths 2a = 2√20 = 4√5

The minor axes of lengths 2b = 2√5.

Compare the equation (x - 4)^2/(√42)^2 + (y + 6)^2/(√23)^2 = 1 with (x - h)^2/a^2 + (y - k)^2/b^2 = 1.

Center = (h, k ) = (4, - 6), a =√42 and b = √23.

The major axes of lengths 2a = 2√42.

The minor axes of lengths 2b = 2√23.