The perpendicular dropped from a point of a circle on a diameter is a mean proportional between the segments into which it divides the diameter.

Consider the equation of the circle with centre at the origin

is given by the equation

Let

be any point on the given circle. Let

be the diameter of the circle as shown in the give diagram. From

draw a perpendicular on diameter

at

. The coordinates of

are

because

lies on the X-axis and

on X-axis. The coordinates of

and

are

and

respectively as shown in diagram.

Now we shall find distance between

and

using distance formula as follows

Also

Now

From equation (i), we have

. Putting this value in equation (ii), we have

This shows that the perpendicular dropped from a point of a circle on a diameter is a mean proportional between the segments into which divides the diameter.

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