# Normal Line of Circle Passes through Origin

The normal lines of a circle pass through the centre of the circle.
Consider the equation of the circle

Since the circle passes through the point $A\left( {{x_1},{y_1}} \right)$ so equation of circle becomes

Now differentiating equation of circle (i) with respect to $x$, we have

If ${m_1}$ is the slope of tangent line at point $A\left( {{x_1},{y_1}} \right)$, then

Slope of normal line at the point $A\left( {{x_1},{y_1}} \right)$ is $m = - \frac{1}{{{m_1}}} = \frac{{{y_1}}}{{{x_1}}}$
Now equation of normal at the point $A\left( {{x_1},{y_1}} \right)$ using slope point is

Now putting the values $x = 0,\,\,y = 0$ in equation (iii), we get the result

This shows that the normal line passes through the centre $\left( {0,0} \right)$ of the circle.