The locus of the centre of the circle`(xcos alpha + y sin alpha - a)^2 + (x sin alpha - y cos alpha - b)^2= k^2` if `alpha` varies, is

Options

A. `x^(2)-y^(2)=a^(2)+b^(2)`
B. `x^(2)-y^(2)=a^(2)b^(2)`
C. `x^(2)+y^(2)=a^(2)+b^(2)`
D. `x^(2)+y^(2)=a^(2)b^(2)`

Answer

3

Explanation

`(x cos alpha+y sing alpha -a)^(2)-(x sin alpha-y cos alpha-b)^(2)=k^(2)`
or `x^(2)+y^(2)+(-2a cos alpha-2b sin alpha)x+(-2a cos alpha+2b sin alpha)y+a^(2)+b^(2)-k^(2)=0`
Centre of the circle is
`(a cos alpha +b sin alpha,a cos alpha-b sin alpha) -= (h,k).`
`h-=a cos alpha +b sin alpha`
`k -= a cos alpha - b sin alpha`
Squaring and adding, we get
`h^(2)+k^(2)=a^(2)+b^(2)`
or `x^(2)+y^(2)=a^(2)+b^(2)`, which is required locus.