The coordinates of the middle point of the chord 

intercepted on line $lx+my+n=0$ by the circle $x^2+y^2=a^2$ are 

Let $\left(x_1,y_1\right)$  e the mid-point of the chord 

 intercepted by the circle  $x^2+y^2=a^2$ on the line $lx+my+n=0$

Then, the equation of the chord of the circle  $x^2+y^2=u^2$

whose middle point is  $\left(x_1,y_1\right)$ is 

$x{x}_1+y{y}_1-a^2=x_1^2+y_1^2-a^2$

$\Rightarrow x{x}_1+y{y}_1=x_1^2+y_1^2 ........\left(i\right)$

Clearly,  $lx+my+n=0$ and (i)    represents the same line.

 $\begin{array}{l}\therefore \frac{x_1}{l}=\frac{y_1}{m}=\frac{-\left(x_1^2+y_1^2\right)}{n}=\lambda ,\text{ say }\\ \Rightarrow x_1=l\lambda ,y_1=m\lambda \text{ and }{x}_1^2+y_1^2=-n\lambda \\ \Rightarrow \left(l^2+m^2\right){\lambda }^2=-n\lambda \\ \Rightarrow \lambda =-\frac{n}{l^2+m^2}\end{array}$

$\therefore x_1=-\frac{li}{l^2+m^2},y_1=\frac{-mm}{l^2+m^2}$

Hence, t e reqmre point  is  $\left(\frac{-m}{l^2+m^2},\frac{-mn}{1^2+m^2}\right)$