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Home » Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.    
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.

    \[{{\mathbf{x}}^{\mathbf{2}}}/\mathbf{25}\text{ }+\text{ }{{\mathbf{y}}^{\mathbf{2}}}/\mathbf{100}\text{ }=\text{ }\mathbf{1}\]

CBSE | Class 11 | Conic Sections | Exercise 11.3 | Maths | NCERT
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.

    \[{{\mathbf{x}}^{\mathbf{2}}}/\mathbf{25}\text{ }+\text{ }{{\mathbf{y}}^{\mathbf{2}}}/\mathbf{100}\text{ }=\text{ }\mathbf{1}\]

Given:

The condition is

    \[{{x}^{2}}/25\text{ }+\text{ }{{y}^{2}}/100\text{ }=\text{ }1\]

Here, the denominator of

    \[{{y}^{2}}/100\]

is more noteworthy than the denominator of

    \[{{x}^{2}}/25\]

.

Thus, the significant pivot is along they-axis, while the minor hub is along thex-axis.

On contrasting the given condition and

    \[~{{x}^{2}}/{{b}^{2}}~+\text{ }{{y}^{2}}/{{a}^{2}}~=\text{ }1\]

, we get

    \[b\text{ }=\text{ }5\text{ }and\text{ }a\text{ }=10.\]

    \[c\text{ }=\text{ }\surd ({{a}^{2}}~\text{ }{{b}^{2}})\]

    \[=\text{ }\surd \left( 100-25 \right)\]

    \[=\text{ }\surd 75\]

    \[=\text{ }5\surd 3\]

Then, at that point,

The directions of the foci are

    \[\left( 0,\text{ }5\surd 3 \right)\text{ }and\text{ }\left( 0,\text{ }-5\surd 3 \right).\]

The directions of the vertices are

    \[\left( 0,\text{ }\surd 10 \right)\text{ }and\text{ }\left( 0,\text{ }-\surd 10 \right).\]

Length of major axis 

    \[=\text{ }2a\text{ }=\text{ }2\text{ }\left( 10 \right)\text{ }=\text{ }20\]

Length of minor axis 

    \[=\text{ }2b\text{ }=\text{ }2\text{ }\left( 5 \right)\text{ }=\text{ }10\]

Eccentricity,

    \[e\text{ }=\text{ }c/a~=\text{ }5\surd 3/10\text{ }=\text{ }\surd 3/2\]

Length of latus rectum

    \[=\text{ }2{{b}^{2}}/a\text{ }=\text{ }(2\times {{5}^{2}})/10\text{ }=\text{ }\left( 2\times 25 \right)/10\text{ }=\text{ }5\]

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