The auxiliary circle of a family of ellipses passes through the origin and makes intercepts of 8 units and 6 units on the x and y-axis, respectively. If the eccentricity of all such ellipses is `1/2,` then find the locus of the focus.

Explanation

`radius = sqrt(4^2 + 3^2) = 5 `

let focus be `(h,k)`

let distanc of focus from centre = al

al = `sqrt((h-4)^2+(k-3)^2)`

a = radius of ac = 5

`(5/2) = sqrt((h-4)^2 + (k-3)^2)`

`(5/2)^2 = (h-4)^2 + (k-3)^2`

`(x-4)^2 + (y-3)^2 = 25/4`