Length of Latus Rectum of Ellipse

The length of 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 ellipse has two foci, so it will have two latus recta.


Let A and B be the ends of the latus rectum as shown in the given diagram. Since latus rectum passes through the focus, so 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}, so 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 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}



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