Question

Find the equation for the ellipse that satisfies the given conditions: 

(i) Vertices (±5,0), foci (±4,0) 

(ii) Vertices (0,±13), foci (0,±5) 

(iii) Vertices (±13,0), foci (±5,0)

Answer 1

(i) Given vertices Vertices (±5, 0), foci (±4,0) 

We have, vertices = (±a, 0) = (±5, 0) a = 5

And foci = (±c, 0) (±4,0) c = 4

We have a²  = b²  + c²

i.e., b²  = a²  - c² = (5)²  -(4)²  = 25 - 16 = 9

i.e., b²  = 9 ⇒ b = ±3

∴ a = 5 and b = 3

∴ Required equation of ellipse is

x²/a² + y²/b² = 1 (∵ foci lie on x -axis)

i.e.,  √(x²/25 + y²/9) = 1

(ii) Given Vertices = (0,±13), foci = (0,±5)

Since foci lie on y-axis, then required equation of the ellipse is

x²/b² + y²/a² = 1 .....(1)

(iii) Given : Vertices = (±13, 0), foci = (±5, 0)

Since foci lie on the x-axis, then required equation of the ellipse is x²/b² + y²/a² = 1

Since, vertices = (±a, 0) = (±13, 0) ∴ a = 13