Question

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

Answer 1

Answer:

$=\frac{x^2}{16} +\frac{y^2}{49}=1$

Step-by-step explanation:

The equation of this ellipse is

$\frac{(x-h)^2}{b^2} +\frac{y-k)^2}{a^2} =1$

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

$a^2-b^2=c^2\\\\b^2=a^2-c^2\\\\\\b^2=7^2-(\sqrt{33} )^2\\\\\\b^2=49-33=16\\\\\\b^2=16$

The equation for the ellipse will be

$\frac{(x-0)^2}{16} +\frac{(y-0)^2}{49} =1\\\\\\=\frac{x^2}{16} +\frac{y^2}{49} =1$