# 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 , the equation of the circle becomes

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

If is the slope of the tangent line at point , then

The slope of the normal line at the point is

Now the equation of the normal at the point using she tlope point is

Now putting the values in equation (iii), we get the result

This shows that the normal line passes through the center of the circle.