Length of the Latus Rectum of an Ellipse

The length of the latus rectum of the ellipse \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1,\,\,a > b is \frac{{2{b^2}}}{a}.

The chord through the focus and perpendicular to the axis of the ellipse is called its latus rectum. Since the ellipse has two foci, it will have two latus recta.


length-of-lr-ellipse

Let A and B be the ends of the latus rectum as shown in the given diagram. Since the latus rectum passes through the focus, abscissa of A and B will be ae. Now putting x = ae in the given equation of ellipse, we have

\begin{gathered} \frac{{{{\left( {ae} \right)}^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1 \\ \Rightarrow \frac{{{y^2}}}{{{b^2}}} = 1 - \frac{{{{\left( {ae} \right)}^2}}}{{{a^2}}} \\ \Rightarrow \frac{{{y^2}}}{{{b^2}}} = \frac{{{a^2} - {{\left( {ae} \right)}^2}}}{{{a^2}}}\,\,\,\,{\text{ - - - }}\left( {\text{i}} \right) \\ \end{gathered}

Since {a^2} - {\left( {ae} \right)^2} = {b^2}, putting this value in equation (i), we have

\frac{{{y^2}}}{{{b^2}}} = \frac{{{b^2}}}{{{a^2}}} \Rightarrow {y^2} = \frac{{{b^4}}}{{{a^2}}} \Rightarrow y = \pm \frac{{{b^2}}}{a}

Thus, A\left( {ae,\frac{{{b^2}}}{a}} \right) and B\left( {ae, - \frac{{{b^2}}}{a}} \right). The length of the latus rectum is

\begin{gathered} l = \left| {AB} \right| = \sqrt {{{\left( {ae - ae} \right)}^2} + {{\left( {\frac{{{b^2}}}{a} - \left( { - \frac{{{b^2}}}{a}} \right)} \right)}^2}} \\ \Rightarrow l = \sqrt {{{\left( {\frac{{2{b^2}}}{a}} \right)}^2}} = \frac{{2{b^2}}}{a} \\ \end{gathered}