Find an equation for the ellipse that satisfies the given conditions. Length of minor axis: 4, foci on y-axis, ellipse passes through the point ( sqrt 2 , sqrt 10 ), centered at the origin

Answer 1

Ellipses are written in the form (x-h)²/a² + (y-k)²/b² = 1

(h, k) is the center, in this case, (0, 0), so we can drop that term and rewrite as

x²/a² + y²/b² = 1

If the foci are on the y-axis, then so is the major axis; as the center is at the origin, the minor axis will then be on the x axis, and a will be 1/2 its length, or 4/2 = 2, and a² = 4

Then, as the ellipse passes through (√2,√10),

x²/a² + y²/b² = 1

(√2)²/4 + (√10)²/b² = 1

2/4 + 10/b² = 1

10/b² = 1/2

b² = 20

Then, the equation of the ellipse is

x²/4 + y²/20 = 1