The perpendicular dropped from the centre of a circle on a chord bisects the chord.

Consider the equation of the circle

Let the points

and

be ends of the chord as shown in the given diagram. Since the circle passes through the point

so equation of circle becomes

Also equation of circle passes through the second point

, so circle becomes

Suppose that

be the midpoint of the chord

, then by using midpoint formula we have

Slope of the chord

is given by

Slope of perpendicular

Equation of line passing through the centre

and perpendicular to the chord is

It is observed that whether the perpendicular (iv) bisects the chord, we check whether

satisfies (iv). Putting the values

in equation (iv), we get

Thus,

satisfies equation (iv), so the perpendicular dropped from the centre of the circle bisects the chord.

Conversely, this can be proves that the perpendicular bisector of any chord of a circle passes through the centre of the circle.

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