Q. The locus of the middle point of the chords of the circle $x^{2} + y^{2} = a^{2}$ such that the chords pass through a given point $\left( x_{1},y_{1} \right)$, is
A) $x^{2} + y^{2} - xx_{1} - yy_{1} = 0$
B) $x^{2} + y^{2} = x_{1}^{2} + y_{1}^{2}$
C) $x + y = x_{1} + y_{1}$
D) $x + y = x_{1}^{2} + y_{1}^{2}$
Explanation
(a)
Let $P\left( x_{1},y_{1} \right)$ be the point, then the chord of contact of tangents drawn from $p$ to the circle $x^{2} + y^{2} = a^{2}$ is $xx_{1} + yy_{1} = a^{2}$
$\therefore x^{2} + y^{2} = a^{2}\left( \frac{xx_{1} + yy_{1}}{a^{2}} \right)$
$\Rightarrow x^{2} + y^{2} - xx_{1} - yy_{1} = 0$
Which is the equation of required locus
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.