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Find the (v) length of the latus rectum of each of the following ellipses.

    \[\mathbf{25}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{4}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{100}\]

CBSE | Class 11 | Ellipse | Maths | RS Aggarwal
Find the (v) length of the latus rectum of each of the following ellipses.

    \[\mathbf{25}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{4}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{100}\]

Given:

    \[\mathbf{25}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{4}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{100}\]

Divide by

    \[100\]

to both the sides, we get

    \[\frac{25}{100}{{x}^{2}}+\frac{4}{100}{{y}^{2}}=1\]

    \[\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{25}=1\]

…(i)

Since,

    \[4\text{ }<\text{ }25\]

So, above equation is of the form,

    \[\frac{{{x}^{2}}}{{{b}^{2}}}+\frac{{{y}^{2}}}{{{a}^{2}}}=1\]

…(ii)

Comparing eq. (i) and (ii), we get

    \[\begin{array}{*{35}{l}}</strong> <strong> {{a}^{2}}=\text{ }25\text{ }and\text{ }{{b}^{2}}=\text{ }4 \\</strong> <strong> \Rightarrow a\text{ }=\text{ }\surd 25\text{ }and\text{ }b\text{ }=\text{ }\surd 4 \\</strong> <strong> \Rightarrow a\text{ }=\text{ }5\text{ }and\text{ }b\text{ }=\text{ }2 \\</strong> <strong>\end{array}\]

(v) To find: Length of the Latus Rectum

We know that,

Length of the Latus Rectum =

    \[\frac{2{{b}^{2}}}{a}\]

    \[<span class="ql-right-eqno"> (1) </span><span class="ql-left-eqno"> </span><img src="https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-81eb80ec2ef0aaa077b5bf8ee6682be1_l3.png" height="149" width="107" class="ql-img-displayed-equation quicklatex-auto-format" alt="\begin{align*} & =\frac{2{{(2)}^{2}}}{5} \\ & =\frac{8}{5} \\ \end{align*}" title="Rendered by QuickLaTeX.com"/>\]

i

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