Question

write an equation for an ellipse centered at the origin, which has foci at (0,±6) and vertices at (0,±sqrt37)

Answer 1

The equation of the ellipse =
`(x^2)/(1) +(y^2)/(37) =1`

Explanation:

Explanation:-

Step(l):-

Given foci of the ellipse is (0,±6)

we know that the foci ( 0, ±C) = (0,±6)

C = 6

The focus is lie on y- axis

Step(ll):-

Given data the vertices are (0,±√37)

The major axes are (0,±a) = (0,±√37)

a = √37

The relation between the focus and semi major axes and semi minor axes are
`c^(2) = a^(2) - b^(2)`

`6^2 = (√(37) )^(2) - b^(2)`

`36 = 37 - b^(2)`

`b^(2) = 37 - 36 =1`

Step (lll) :-

The equation of the ellipse
`(x^2)/(b^2) +(y^2)/(a^(2) ) =1`

`(x^2)/(1^2) +\frac{y^2}{\sqrt({37} )^(2) } =1`

`(x^2)/(1) +(y^2)/(37) =1`

Conclusion:-

The equation of the ellipse =
`(x^2)/(1) +(y^2)/(37) =1`