If the chord of contact of tangents from a point `P` to a given circle passes through `Q ,` then the circle on `P Q` as diameter. cuts the given circle orthogonally touches the given circle externally touches the given circle internally none of these

Options

A. cuts the given circle orthogonally
B. touches the given circle externally
C. touches the given circle internally
D. none of these

Answer

A

Explanation

Let `P(x_(1), y_(1))` and `Q(x_(2), y_(2))` be the given points and `x^(2)+y^(2)=a^(2)` be the circle. The chord of contact of tangents drawn from `P(x_(1), y_(1))` to `x^(2)+y^(2)=a^(2)` is
`x x_(1)+y y_(1)=a^(2)`
If it passes through `Q(x_(2), y_(2))`, then
`x_(1) x_(2)+y_(1)y_(2)=a^(2) " " ...(i)`
The equation of the circle on PQ as diameter is
`(x-x_(1))(x-x_(2))+(y-y_(1))(y-y_(2))=0`
`rArr x^(2)+y^(2)-x(x_(1)+x_(2))-y(y_(1)+y_(2))+x_(1)x_(2)+y_(1)y_(2)=0`
This circle will cut the given circle orthogonally, if
`0(x_(1)+x_(2))+0(y_(1)+y_(2))=a^(2)+x_(1)x_(2)+y_(1)y_(2)`
`rArr x_(1)x_(2)+y_(1)y_(2)=a^(2)=0`, which is true. [Using (i)]