Geo5.2.44

Find the equation of the ellipse whose vertices are $(-4,1),(6,1)\,$ and whose focus lies on the line $x-2y=2\,$

Given vertices $(-4,1),(6,1)\,$

The distance between the vertices is $10=2a,a=5\,$

Therefore centre= $\left(\frac{-4+6}{2},\frac{1+1}{2}\right)\,$

Equation to the major axis is $y-1=0\,$

Focus lies on the line $x-2y=2\,$

Solving these two equations, we get the point as (4,1).

The distance from the centre to the focus is ae=3

Hence $a^2 e^2=a^2-b^2,b^2=16\,$

Hence the equation of ellipse is $\frac{(x-1)^2}{25}+\frac{(y-1)^2}{16}=1\,$