Question

Question #50910

Answer 1

In the given problem the ellipse has got coordinates of

`"foci"->(1, 5) ;(1, -1);` `"center"-> (1, 2)`
`"vertices"-> (1, 6); (1, -2) (-1.65, 2);(3.65, 2)`

Here coordinates of foci,center and vertices have got same abscissa `1`. So the major axis of the ellipse is parallel to y-axis and it's equation is `y=1`

Hence the equation of the ellipse will be of following standard form

`color(blue)((x-alpha)^2/b^2+(y-beta)^2/a^2)=1`,

Where coordinates of the center are `(alpha,beta)` and length of semi-major axis is `a` and that of semi-minor axis is `b`.

So we have `alpha=1 and beta=2`

`a=(6-2)=4`

The coordinates of the end points of semi-minor axis are `(-1.65, 2);(3.65, 2)`. So the length of semi-minor axis `b=(3.65-1)=2.65=>b^2=2.65^2~~7`

Again we know that distance between center and focus

`=asqrt(1-b^2/a^2)=(5-2)=3`

`=>a^2(1-b^2/a^2)=9`

`=>a^2-b^2=9`

`=>4^2-b^2=9`

`=>b^2=7`

Hence the equation of ellipse is

`color(red)((x-1)^2/(sqrt7)^2+(y-2)^2/4^2=1)`,

`=>color(red)((x-1)^2/7+(y-2)^2/16=1)`,