Given:
The condition is
![\[~9{{y}^{2}}~\text{ }4{{x}^{2}}~=\text{ }36\text{ }or\text{ }{{y}^{2}}/4\text{ }\text{ }{{x}^{2}}/9\text{ }=\text{ }1\text{ }or\text{ }{{y}^{2}}/{{2}^{2}}~\text{ }{{x}^{2}}/{{3}^{2}}~=\text{ }1\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-91356216f3cd9d00c00cc5bbdc32b845_l3.png "Rendered by QuickLaTeX.com")
On contrasting this condition and the standard condition of hyperbola
![\[{{y}^{2}}/{{a}^{2}}~\text{ }{{x}^{2}}/{{b}^{2}}~=\text{ }1,\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a87b8bdf815dd19b5a6529f1fc3f7fd5_l3.png "Rendered by QuickLaTeX.com")
We get
![\[a\text{ }=\text{ }2\text{ }and\text{ }b\text{ }=\text{ }3,\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-3ff6cc2dedc55c9ae09458679f885d8e_l3.png "Rendered by QuickLaTeX.com")
It is realized that,
![\[{{a}^{2}}~+\text{ }{{b}^{2}}~=\text{ }{{c}^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9d5cafbce671459336f5a04274292a61_l3.png "Rendered by QuickLaTeX.com")
So,
![\[\begin{array}{*{35}{l}} {{c}^{2}}~=\text{ }4~+\text{ }9 \\ {{c}^{2}}~=\text{ }13 \\ c\text{ }=~\surd 13 \\ Then, \\ The\text{ }coordinates\text{ }of\text{ }the\text{ }foci\text{ }are\text{ }\left( 0,~\surd 13 \right)\text{ }and\text{ }\left( 0,\text{ }\surd 13 \right). \\ The\text{ }coordinates\text{ }of\text{ }the\text{ }vertices\text{ }are\text{ }\left( 0,\text{ }2 \right)\text{ }and\text{ }\left( 0,\text{ }\text{ }2 \right). \\ Eccentricity,\text{ }e\text{ }=\text{ }c/a\text{ }=~\surd 13/2 \\ Length\text{ }of\text{ }latus\text{ }rectum\text{ }=\text{ }2{{b}^{2}}/a\text{ }=\text{ }\left( 2\text{ }\times \text{ }{{3}^{2}} \right)/2\text{ }=\text{ }\left( 2\times 9 \right)/2\text{ }=\text{ }18/2\text{ }=\text{ }9 \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-32a57e0503ee2485565d3c294f957e7d_l3.png "Rendered by QuickLaTeX.com")
Given:
The condition is
On contrasting this condition and the standard condition of hyperbola
We get
It is realized that,
So,
