The locus of the middle points of chords of the circle $x^2+y^2=a^2$ which are parallel to the line $y=mx$ is

Let $(x_1,y_1)$  be the mid-point of the chord

$S\equiv x^2+y^2-a^2$

$\therefore$ The equation of the chord is $S_1=S_{11}$

$x{x}_1+y{y}_1=x_1^2+y_1^2\to \left(1\right)$

Slope of (1) is $=\frac{-x_1}{y_1}=m_1$

$mx-y=0\to \left(2\right)\Rightarrow$ Slope of (2) $=\frac{-m}{-1}=m=m_2$

Since (1) and (2) are parallel $\Rightarrow \frac{-x_1}{y_1}=m\Rightarrow x+my=0$

$\Rightarrow \frac{-x_1}{y_1}=m\Rightarrow x+my=0$

$\therefore$ The locus of $p(x_1,y_1)$  is $x+my=0$