write the standard form of conic equation

Question

Write the standard form of the conic equation in which Eccentricity = sqr(11)/4

Vertices:  (0, 9+4(sqr 5)), (0, 9-4(sqr(5))

 

I have the answer of x^2  /25 + (y-9)^2  /80 = 1 but I don't understand how to get it.  Thank you so much.

Answer 1

Eccentricity= c/a=sqrt(11)/4

 

As per given vertices,major axis is vertical.

 

9+4sqrt(5)-9+4sqrt(5)=8sqrt(5)=2a

So,    a=4sqrt(5)

 

c/a=sqrt(11)/4. =c/4sqrt(5)

 

c=sqrt(55)

 

c^2= a^2. -b^2

 

55=80-b^2

b=5

 

Center coordinates=   (0,9)

 

Equation of ellipse is

 

[(x^2)/25]  + [(y-9)^2]/80=1

 

YOU are welcome to revert back if any step is not clear.