# 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.

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.