Question

Find the standard equation for the ellipse, using the given characteristic or characteristics. vertices:(0,+-7) foci: (0,+-√33)

Answer 1

`=(x^2)/(16) +(y^2)/(49)=1`

Explanation:

The equation of this ellipse is

`((x-h)^2)/(b^2) +(y-k)^2)/(a^2) =1`

for a vertical oriented ellipse where;

(h,k) is the center

c=distance from center to the foci

a=distance from center to the vertices

b=distance from center to the co-vertices

You know center of an ellipse is half way between the vertices , hence the center (h,k) of this ellipse is (0,0) and its is vertical oriented ellipse

Given that

a= distance between the center and the vertices, a=7

c=distance between the center and the foci, c=√33

Then find b

`a^2-b^2=c^2\\\\b^2=a^2-c^2\\\\\\b^2=7^2-(√(33) )^2\\\\\\b^2=49-33=16\\\\\\b^2=16`

The equation for the ellipse will be

`((x-0)^2)/(16) +((y-0)^2)/(49) =1\\\\\\=(x^2)/(16) +(y^2)/(49) =1`