Question

How do you write an equation of an ellipse in standard form given vertices (-5, 4) and (8, 4) and whose focus is (-4, 4)?

Answer 1

`color(blue)((2x-3)^2/169+(y-4)^2/12=1)`

Explanation:

Given that the the coordinates of vertices of the ellipse are

`"Vertices"->(-5,4) " & "(8,4)`

The ordinates of vertices being same (4)the axis of the ellipse is parallel to x-axis.

`"Center"->((-5+8)/2,(4+4)/2)->(1.5,4)`

*If a and b are halves of the major and minor axis respectively then the standard equation of ellipse may be written as

#color(red)((x-1.5)^2/a^2+(y-4)^2/b^2=1)..... (1)#

Now we are to findout a and b.

Again it is also given the coordinate of

`"Focus"->(-4,4)`

Now a is the distance between center and vertex.

`a=sqrt((8-1.5)^2+(4-4)^2)=6.5`

Now if e represnts eccentricity of the ellipse then

`e=sqrt((a^2-b^2)/a^2)`

Now the distance between center and focus is ae

`:.ae=sqrt((-1.5-4)^2+(4-4)^2)=5.5`

`=>a^2e^2=5.5^2=>(a^2(a^2-b^2))/a^2=5.5^2`

`=>6.5^2-b^2=5.5^2`

`=>b^2=12`

Now inserting the value of a and b in equation (1) we get the equation of ellipse as

`color(blue)((x-1.5)^2/6.5^2+(y-4)^2/12=1)`

`color(blue)(=>(2x-3)^2/169+(y-4)^2/12=1)`