# Normal Line of a Circle Passes through the Origin

The normal lines of a circle passes through the center of the circle.

Consider the equation of the circle

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

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

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

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

Now the equation of the normal at the point $A\left( {{x_1},{y_1}} \right)$ using she tlope 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 center $\left( {0,0} \right)$ of the circle.