(i)
![\[4{{x}^{2}}-\text{ }3{{y}^{2}}~=\text{ }36\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-826602d6f6c79c29cf01638fb20ce3d4_l3.png "Rendered by QuickLaTeX.com")
Given:
The equation
![\[=>\text{ }4{{x}^{2}}-\text{ }3{{y}^{2}}~=\text{ }36\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-534eec6647bd4cd9388e76b39f0a79ee_l3.png "Rendered by QuickLaTeX.com")
The equation can be expressed as:
The obtained equation is of the form
Where,
![\[{{a}^{2}}~=\text{ }9,\text{ }{{b}^{2}}~=\text{ }12\text{ }i.e.,\text{ }a\text{ }=\text{ }3\text{ }and\text{ }b\text{ }=\text{ }\surd 12\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-f2e19fe9e7dd0a9569ef4ead74f67d38_l3.png "Rendered by QuickLaTeX.com")
Eccentricity is given by:
Foci: The coordinates of the foci are
![\[\left( \pm ae,\text{ }0 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d3cb528ebfdb03c275158177df0f88e4_l3.png "Rendered by QuickLaTeX.com")
![\[\left( \pm ae,\text{ }0 \right)\text{ }=\text{ }\left( \pm \surd 21,\text{ }0 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5a5867de5ad7c5aee78076e021657b9f_l3.png "Rendered by QuickLaTeX.com")
The equation of directrices is given as:
The length of latus-rectum is given as:
![\[2{{b}^{2}}/a\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a3ecfe1d612a1e0e995bd1475f8f34e4_l3.png "Rendered by QuickLaTeX.com")
![\[=\text{ }2\left( 12 \right)/3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0e74ab987dd469c567d0e3482e31991f_l3.png "Rendered by QuickLaTeX.com")
Or,
![\[=\text{ }24/3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-f001fbd53a3cc84e1e29011e3a72799c_l3.png "Rendered by QuickLaTeX.com")
![\[=\text{ }8\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-bc58a03f20fb2cbf59dc3263a0e2d84a_l3.png "Rendered by QuickLaTeX.com")
(ii)
![\[3{{x}^{2}}-\text{ }{{y}^{2}}~=\text{ }4\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c2722c48d901f3463bf92d0054f61491_l3.png "Rendered by QuickLaTeX.com")
Given:
The equation
![\[=>\text{ }3{{x}^{2}}-\text{ }{{y}^{2}}~=\text{ }4\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-43c79d1bc107d251113019d3dc97bf00_l3.png "Rendered by QuickLaTeX.com")
The equation can be expressed as:
The obtained equation is of the form
Where,
![\[a\text{ }=\text{ }2/\surd 3\text{ }and\text{ }b\text{ }=\text{ }2\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-546bfa1ee2b4d6795962bfd3a0d7b08e_l3.png "Rendered by QuickLaTeX.com")
Eccentricity is given by:
Foci: The coordinates of the foci are
![\[\left( \pm ae,\text{ }0 \right)\text{ }=\text{ }\pm \left( 2/\surd 3 \right)\left( 2 \right)\text{ }=\text{ }\pm 4/\surd 3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-403d097edd51d7ca94179b9961b2468a_l3.png "Rendered by QuickLaTeX.com")
![\[\left( \pm ae,\text{ }0 \right)\text{ }=\text{ }\left( \pm 4/\surd 3,\text{ }0 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-dfecd0a980c84dc8f1c90237bd97fa66_l3.png "Rendered by QuickLaTeX.com")
The equation of directrices is given as:
The length of latus-rectum is given as:
![\[=\text{ }2\left( 4 \right)/\left[ 2/\surd 3 \right]\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-47131811d62286f352dea31b73727063_l3.png "Rendered by QuickLaTeX.com")
So,
![\[=\text{ }4\surd 3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-846456cb441aa647462b41b3ae587de3_l3.png "Rendered by QuickLaTeX.com")