Question

How do you write an equation of an ellipse in standard form given eccentricity of 0.5 if the vertices for the ellipse occur at the points ( 0 , 7 ) and ( 0, -9 )?

Answer 1

Given that the vertices of an ellipse are (0,7) and (0,-9).
We can say that the center of the ellipse is `(0","(7-9)/2)=(0,-1)` and its major axis is on the y-axis. The length of major axis `2a=7-(-9)=16`
and semi major axis `a=8`

Now it is given that its eccentricity `(e)=0.5`

Again we know

`e^2=(a^2-b^2)/a^2`,where b is the length of semi minor axis.

So `(0.5)^2=(8^2-b^2)/8^2`

`=>1/4=1-b^2/64`

`=>b^2=64xx(1-1/4)=48`

`->b=4sqrt3`

So standard equation of the ellipse having center (0,-1) and semi major axis a=8 (along y-axis) and semi minor axis `b= 4sqrt3` (parallel to xaxis )is given by

`(x-0)^2/(4sqrt3)^2+(y-(-1))^2/8^2=1`

`=>x^2/(4sqrt3)^2+(y+1)^2/8^2=1`

`=>x^2/48+(y+1)^2/64=1`