Question

What is the ratio of the greatest and least focal distances ofa point on the ellipse `4x^(2)+9y^(2)=36?`

Answer 1

Correct Answer - `(7+3sqrt(5))/(2)`
Given ellipse is `(x^(2))/(9)+(y^(2))/(4)=1(agtb)`.
`a^(2)=9,b^(2)=4,e=sqrt((a^(2)-b^(2))/(a^(2)))=sqrt((9-4)/(9))=sqrt(5)/(4)`
Greatest focal distane a=+ae
Least focal distance =a-ae
`:.` Required ratio `(a+ae)/(a-ae)+(1+e)/(a-e)`
`=(1+(sqrt(5))/(3))/(1-(sqrt(5))/(3))=(3+sqrt(5))/(3+sqrt(5))`
`=((3+sqrt(5))^(2))/(4)=(7+3sqrt(5))/(2)`