Perpendicular from any Point of a Circle on a Diameter is a Mean Proportional

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.


diameter-mean-proportional

Consider the equation of the circle with a center at the origin O\left( {0,0} \right) is given by the equation

{x^2} + {y^2} = {r^2}\,\,\,\,{\text{ - - - }}\left( {\text{i}} \right)

Let P\left( {x,y} \right) be any point on the given circle. Let AB be the diameter of the circle as shown in the given diagram. From P draw a perpendicular on diameter AB at M. The coordinates of M are M\left( {x,0} \right) because M lies on the X-axis and y = 0 lies on the X-axis. The coordinates of A and B are A\left( { - a,0} \right) and B\left( {a,0} \right) respectively as shown in diagram.

Now we shall find the distance between AM and MB using the distance formula as follows:

\begin{gathered} \left| {AM} \right| = \sqrt {{{\left( { - a - x} \right)}^2} + {{\left( {0 - 0} \right)}^2}} \\ \,\,\,\,\,\,\,\,\,\,\,\,\, = \sqrt {{{\left( {a + x} \right)}^2} + 0} = \sqrt {{{\left( {a + x} \right)}^2}} = a + x \\ \end{gathered}

Also

\begin{gathered} \left| {MB} \right| = \sqrt {{{\left( {a - x} \right)}^2} + {{\left( {0 - 0} \right)}^2}} \\ \,\,\,\,\,\,\,\,\,\,\,\,\, = \sqrt {{{\left( {a - x} \right)}^2} + 0} = \sqrt {{{\left( {a - x} \right)}^2}} = a - x \\ \end{gathered}

Now

 \Rightarrow \left| {AM} \right|\left| {MB} \right| = \left( {a + x} \right)\left( {a - x} \right) = {a^2} - {x^2}\,\,\,\,\,\,\,\,\,\,\left( {\because a > x} \right)\,\,\,{\text{ - - - }}\left( {{\text{ii}}} \right)

From equation (i), we have {y^2} = {r^2} - {x^2}. Putting this value in equation (ii), we have

\begin{gathered} \left| {AM} \right|\left| {MB} \right| = {y^2} \\ \Rightarrow \left| {AM} \right|\left| {MB} \right| = {\left| {PM} \right|^2} \\ \end{gathered}

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