Two tangents are drawn from any point P on a given line `L=0` to a given circle `S=0`, if these tangents touch the circle at point A and B, then locus of the circumcentre the `DeltaPAB` will be

Options

A. A pair of straight lines
B. A circle touching the given line `L=0`
C. A circle touching the given circle `S=0`
D. A line parallel to given line `L=0`

Answer

D

Explanation

The circumcirle of `DeltaPAB` passes through C with PC as its diameter. Therefore centre of this circle `(C_(1))` will be mid pt of PC.
Let `L=ax+by+c=0`
`P(x_(1).y_(1)) & C(p,q)`
`C_(1)(h,k)`
`h=(x_(1)+p)/(2)`
`k=(y_(1)+q)/(2)`
`p(x_(1),y_(1))` passes through `L=0`
`impliesax_(1)+by_(1)+c=0`
or `a(2h-p)+b(2k-q)+c=0`
or `2ah+2bk-ap-bq+c=0`
Locus of `(h,k)` is `ax+by=(ap+bq+c)/(2)`
`implies` A line parallel to `L=0`