# 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.