(i) the distance between the
![\[foci\text{ }=\text{ }16\text{ }and\text{ }eccentricity\text{ }=\text{ }\surd 2\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-becf7b6bb4c2c2bf45a8b50bf4365f2d_l3.png "Rendered by QuickLaTeX.com")
Given:
Distance between the foci
![\[=\text{ }16\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7aba7516a87b5523fb1697be6bc231d6_l3.png "Rendered by QuickLaTeX.com")
Eccentricity
![\[=\text{ }\surd 2\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-92fc478ccdafb014e8575ee0a131b641_l3.png "Rendered by QuickLaTeX.com")
Let us compare with the equation of the form
Distance between the foci is
![\[2ae\text{ }and\text{ }{{b}^{2}}~=\text{ }{{a}^{2}}({{e}^{2}}-\text{ }1)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6a28cb6f033aa289b7d3c5910982f2b3_l3.png "Rendered by QuickLaTeX.com")
So,
![\[2ae\text{ }=\text{ }16\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e52653aa48a93afde9696dc1fcf8b2b6_l3.png "Rendered by QuickLaTeX.com")
![\[ae\text{ }=\text{ }16/2\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-114b5690aff1281279da6e8dd32df68a_l3.png "Rendered by QuickLaTeX.com")
Or,
![\[a\surd 2\text{ }=\text{ }8\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-89a51977d7159663b78b6778e5f29a2e_l3.png "Rendered by QuickLaTeX.com")
![\[a\text{ }=\text{ }8/\surd 2\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5070f18c2c85957da588eec32f6c317e_l3.png "Rendered by QuickLaTeX.com")
Or,
![\[{{a}^{2}}~=\text{ }64/2\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1da2a4ba39c88bd5bee0e7c219c7d595_l3.png "Rendered by QuickLaTeX.com")
![\[=\text{ }32\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0dc1ae6049e1a7825b0bd8fe7d8e304e_l3.png "Rendered by QuickLaTeX.com")
We know that,
![\[{{b}^{2}}~=\text{ }{{a}^{2}}({{e}^{2}}-\text{ }1)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-66737f3b0162f3b7f54a517ccd600f7f_l3.png "Rendered by QuickLaTeX.com")
So,
![\[{{b}^{2}}~=\text{ }32\text{ }[{{\left( \surd 2 \right)}^{2}}-\text{ }1]\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c5fc907a461696f970d0016a75d5ad9c_l3.png "Rendered by QuickLaTeX.com")
![\[=\text{ }32\text{ }\left( 2\text{ }-\text{ }1 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-74baf753e992993d4c13b17328037c8a_l3.png "Rendered by QuickLaTeX.com")
Hence,
The Equation of hyperbola is given as
![\[{{x}^{2}}-\text{ }{{y}^{2}}~=\text{ }32\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-dad1f75cc6cf346f04195574a85a8e6f_l3.png "Rendered by QuickLaTeX.com")
∴ The Equation of hyperbola is
(ii) conjugate axis is
![\[5\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7dc8465919d0838925dbc6c99670258c_l3.png "Rendered by QuickLaTeX.com")
and the distance between foci
![\[=\text{ }13\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c83e968add4833e03486b314c82bd63e_l3.png "Rendered by QuickLaTeX.com")
Given:
Conjugate axis
![\[=\text{ }5\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e45ddd13b94c34a71f1a55ef7d8c9ef2_l3.png "Rendered by QuickLaTeX.com")
Distance between foci
Let us compare with the equation of the form
Distance between the foci is
Length of conjugate axis is
![\[2b\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-fc11e509db8bd1a945da3d9a3f24658b_l3.png "Rendered by QuickLaTeX.com")
So,
![\[2b\text{ }=\text{ }5\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-46b38944aa2b6b20735e4bff70482b83_l3.png "Rendered by QuickLaTeX.com")
![\[b\text{ }=\text{ }5/2\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-107e74ed281d01e6a99e40d621b92df7_l3.png "Rendered by QuickLaTeX.com")
Or,
![\[{{b}^{2}}~=\text{ }25/4\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-241853dae33bc5a71a831d00cf0b52ac_l3.png "Rendered by QuickLaTeX.com")
We know that,
![\[2ae\text{ }=\text{ }13\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1d5c6cddcca8b77fb2c7ffdda26fc18d_l3.png "Rendered by QuickLaTeX.com")
![\[ae\text{ }=\text{ }13/2\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a83dd634094159f543f1de22e24f0a93_l3.png "Rendered by QuickLaTeX.com")
Or,
![\[{{a}^{2}}{{e}^{2}}~=\text{ }169/4\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-f38c4f24acabc99f59bb5b114d1d20ac_l3.png "Rendered by QuickLaTeX.com")
Or,
![\[{{b}^{2}}~=\text{ }{{a}^{2}}{{e}^{2}}-\text{ }{{a}^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c8e6017570999179233a7e1b2a1bc892_l3.png "Rendered by QuickLaTeX.com")
![\[25/4\text{ }=\text{ }169/4\text{ }-\text{ }{{a}^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-03de636914277d908abf2200bf77e9e1_l3.png "Rendered by QuickLaTeX.com")
Or,
![\[{{a}^{2}}~=\text{ }169/4\text{ }-\text{ }25/4\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-26093f3d91205ddee9ccbab71f65de37_l3.png "Rendered by QuickLaTeX.com")
![\[=\text{ }144/4\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5bb367f1e8a098580a1d4468edf8027d_l3.png "Rendered by QuickLaTeX.com")
So,
![\[=\text{ }36\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-295f12702317ed6549b3af74ebc0f0a6_l3.png "Rendered by QuickLaTeX.com")
The Equation of hyperbola is given as
∴ The Equation of hyperbola is
![\[25{{x}^{2}}~\text{ }144{{y}^{2}}~=\text{ }900\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1685c5946413c1bd22468b318b5829e2_l3.png "Rendered by QuickLaTeX.com")
