Question

Find the equation of an ellipse whose focii are `(-1,-3)` and `(-1,21)` and major axis is `30`?

Answer 1

Equation of ellipse is `(x+1)^2/9^2+(y-9)^2/15^2=1`

Explanation:

As focii are `(-1,-3)` and `(-1, 21)`, the center of ellipse midpoint of focii and is `(-1,(-3+21)/2)` or `(-1,9)` and major axis, which is `30`, is parallel to `y`-axis.

If major axis `2a=30`, `a=15` and `ae=(21-(-3))/2=12`, where `e` is eccentricity and `e=sqrt(1-b^2/a^2)=12/15=4/5`

or `1-b^2/a^2=16/25` i.e. `b^2/a^2=9/25` and `b=3/5xxa=9`

Hence, the equation is

`(x+1)^2/9^2+(y-9)^2/15^2=1`

graph{((x+1)^2/9^2+(y-9)^2/15^2-1)((x+1)^2+(y+3)^2-0.5)((x+1)^2+(y-21)^2-0.5)=0 [-44.84, 35.16, -8.64, 31.36]}