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.

Options

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

Answer

1

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 its passes through `Q(x_(2),y_(2))`, then
`x_(1)x_(2) +y _(1)y_(2)=a^(2)`
The equation of the circle on PQ as diameter is
`(x-x_(1))(x-x_(2))+(y-y_(1))(y-y_(2))=0`
or `x^(2)+y^(2)-x9x_(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+0= -a^(2)+x_(1)x_(2)+y_(1)y_(2)`
or `x_(1)x_(2)+y_(1)y_(2)-a^(2)=0`
which is true by (1).