Question

If $P,Q$ is a pair of conjugate points with respect to a circle $S$, then the circle on $PQ$ as diameter

• A. Touches the circle $S$
• B. Cuts the circle $S$ at an angle $\frac{\pi }{4}$
• C. Cuts the circle $S$ orthogonally
• D. None of these

Answer 1

Answer: (C)

Cuts the circle $S$ orthogonally

Let the coordinates of $P$ and $Q$ be $(x_1,y_1)$ and $(x_2,y_2),$ respectively.

Let the given circle be $S\equiv x^2+y^2-a^2=0$ ...(1)

Polar of $P$ w.r.t. (1) is ${xx}_1+y{y}_1=a^2.$

$\because P,Q$ are conjugate points,

$\therefore$ polar of $P$ passes through $Q$

$\Rightarrow x_1{x}_2+y_1{y}_2=a^2$ ...(2)

Equation of circle on $PQ$ as diameter is $(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0$

or, $x^2+y^2-(x_1+x_2)x-(y_1+y_2)y+(x_1{x}_2+y_1{y}_2)=0$

Using (2)

$x^2+y^2-(x_1+x_2)x-(y_1+y_2)y+a^2=0$ ...(3)

Clearly, circles (1) and (3) cut each other orthogonally.