Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. eccentricity 1 2 , vertices on the x-axis, passing through (1, 3)

Question

Find an equation for the ellipse that has its center at the origin and satisfies the given conditions.

eccentricity 1

2

,    vertices on the x-axis,    passing through (1, 3)

Answer 1

b^2 = (1-e^2)a^2 = (1-1/4)a^2 = 3/4a^2

Then, as the ellipse passed through (1, 3) and is centered at the origin

x^2/a^2 + y^2/b^2 = 1

1^2/a^2 + 3^2/(3/4a^2) = 1

13/a^2 = 1

a^2 = 13

b^2 = 3/4a^2 = 39/4

x^2/13 + 4y^2/39 = 1