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.