#### The coordinates of the middle point of the chord intercepted on line $\mathrm{l x+m y+n=0}$ by the circle $\mathrm{x^2+y^2=a^2}$ areOption: 1 Option: 2 Option: 3 Option: 4

Let $\mathrm{\left(x_1, y_1\right)}$ be the mid-point of the chord intercepted by the circle $\mathrm{x^2+y^2=a^2}$ on the line $\mathrm{l x+m y+n=0.}$

Then, the equation of the chord of the circle $\mathrm{x^2+y^2=a^2}$ whose middle point is $\mathrm{\left(x_1, y_1\right)}$ is

\mathrm{ \begin{aligned} & 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 \end{aligned} }                  $..(i)$

Clearly, $\mathrm{l x+m y+n=0}$ and (i) represents the same line.

\mathrm{ \begin{aligned} & \therefore \quad \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 \quad\left(l^2+m^2\right) \lambda^2=-n \lambda \Rightarrow \lambda=-\frac{n}{l^2+m^2} \\\\ & \therefore \quad x_1=-\frac{l n}{l^2+m^2}, y_1=\frac{-m n}{l^2+m^2} \end{aligned} }

Hence, the required point is $\mathrm{\left(\frac{-l n}{l^2+m^2}, \frac{-m n}{l^2+m^2}\right)}$