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Home » Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas. 9y^2 – 4x^2 = 36
Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas. 9y^2 – 4x^2 = 36
CBSE | Class 11 | Conic Sections | Exercise 11.4 | Maths | NCERT
Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas. 9y^2 – 4x^2 = 36

Given:

The condition is

    \[~9{{y}^{2}}~\text{ }4{{x}^{2}}~=\text{ }36\text{ }or\text{ }{{y}^{2}}/4\text{ }\text{ }{{x}^{2}}/9\text{ }=\text{ }1\text{ }or\text{ }{{y}^{2}}/{{2}^{2}}~\text{ }{{x}^{2}}/{{3}^{2}}~=\text{ }1\]

On contrasting this condition and the standard condition of hyperbola

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

We get

    \[a\text{ }=\text{ }2\text{ }and\text{ }b\text{ }=\text{ }3,\]

It is realized that,

    \[{{a}^{2}}~+\text{ }{{b}^{2}}~=\text{ }{{c}^{2}}\]

So,

    \[\begin{array}{*{35}{l}} {{c}^{2}}~=\text{ }4~+\text{ }9 \\ {{c}^{2}}~=\text{ }13 \\ c\text{ }=~\surd 13 \\ Then, \\ The\text{ }coordinates\text{ }of\text{ }the\text{ }foci\text{ }are\text{ }\left( 0,~\surd 13 \right)\text{ }and\text{ }\left( 0,\text{ }\surd 13 \right). \\ The\text{ }coordinates\text{ }of\text{ }the\text{ }vertices\text{ }are\text{ }\left( 0,\text{ }2 \right)\text{ }and\text{ }\left( 0,\text{ }\text{ }2 \right). \\ Eccentricity,\text{ }e\text{ }=\text{ }c/a\text{ }=~\surd 13/2 \\ Length\text{ }of\text{ }latus\text{ }rectum\text{ }=\text{ }2{{b}^{2}}/a\text{ }=\text{ }\left( 2\text{ }\times \text{ }{{3}^{2}} \right)/2\text{ }=\text{ }\left( 2\times 9 \right)/2\text{ }=\text{ }18/2\text{ }=\text{ }9 \\ \end{array}\]

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