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

The auxiliary circle makes intercepts of 8 units and 6 units on he x-and y-axes, respectively. Therefore, the centre of the circle is (4,3) and the radius is 10.
(4,3) is the centre of inscirbed ellipse also.
Now, the distance ofteh focus the from the centre of the ellipse is ae.
Since eh radius of the auxiliary circle is 5, the length of the semimajor axis, a will be 5 .
Given `e=(1)/(2)`
`:. ae=(5)/(2)`
Therefore, the locus of the focus is
`(x-4)^(2)+(y-3)^(2)=(25)/(4)`