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

so equation of circle becomes

Now differentiating equation of circle (i) with respect to

, we have

If

is the slope of tangent line at point

, then

Slope of normal line at the point

is

Now equation of normal at the point

using slope point is

Now putting the values

in equation (iii), we get the result

This shows that the normal line passes through the centre

of the circle.

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