Find the locus of the midpoint of the chord of the circle `x^2+y^2-2x-2y=0` , which makes an angle of `120^0` at the center.

Answer

`(x-1)^(2)+(y-1)^(2)=1`

Explanation

Given circles is `(x-1)^(2)+(y-1)^(2)=4`.

In the figure, chord AB subtends an angle of `120^(@)` at the centre . M (h,k) is the midpoint of chord AB.
In right angled triangle AMC,
`CM =AC cos 60^(@)`
`sqrt((h-1)^(2)+(k-1)^(2))=2 xx (1)/(2)`
Therefore, required locus is `(x-1)^(2)+(y-1)^(2)=1`