![\[{{x}^{2}}-\text{ }3{{y}^{2}}-\text{ }2x\text{ }=\text{ }8\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-06452125b9efb66f6925782f60a837ed_l3.png "Rendered by QuickLaTeX.com")
Given:
The equation
![\[=>\text{ }{{x}^{2}}-\text{ }3{{y}^{2}}-\text{ }2x\text{ }=\text{ }8\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6dec1c08dfe1833d10f14bf920131770_l3.png "Rendered by QuickLaTeX.com")
Let us find the centre, eccentricity, foci and directions of the hyperbola
By using the given equation
![\[{{x}^{2}}-\text{ }2x\text{ }+\text{ }1\text{ }-\text{ }3{{y}^{2}}-\text{ }1\text{ }=\text{ }8\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-644382f029ae33a9cdf832b73bbb6fae_l3.png "Rendered by QuickLaTeX.com")
So,
![\[{{\left( x\text{ }-\text{ }1 \right)}^{2}}-\text{ }3{{y}^{2}}~=\text{ }9\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-f0d0b15e8280e126a66bc42551bfc66f_l3.png "Rendered by QuickLaTeX.com")
Here, center of the hyperbola is
![\[\left( 1,\text{ }0 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b2eb035cda94911f28410edc36dd71d8_l3.png "Rendered by QuickLaTeX.com")
So, let
![\[x\text{ }-\text{ }1\text{ }=\text{ }X\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c18e725da32f35defd0a32ed732aa16f_l3.png "Rendered by QuickLaTeX.com")
The obtained equation is of the form
Where,
![\[a\text{ }=\text{ }3\text{ }and\text{ }b\text{ }=\text{ }\surd 3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6574590e06fcd731b0c6322db9cb5cd1_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")
![\[X\text{ }=~\pm ~2\surd 3\text{ }and\text{ }Y\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-3d26427016b6c27d168590864d7caf1f_l3.png "Rendered by QuickLaTeX.com")
![\[X\text{ }-\text{ }1\text{ }=~\pm ~2\surd 3\text{ }and\text{ }Y\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-8b976dbf13c5ac53b355826a2dc79b69_l3.png "Rendered by QuickLaTeX.com")
So,
![\[X=~\pm ~2\surd 3\text{ }+\text{ }1\text{ }and\text{ }Y\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-f3721984a4d0c7203d655b88b39b05d8_l3.png "Rendered by QuickLaTeX.com")
So,
![\[Foci\text{ }=\text{ }\left( 1~\pm ~2\surd 3,\text{ }0 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-52e24745befbc9dba846c5ed0e01f41b_l3.png "Rendered by QuickLaTeX.com")
Equation of directrix are:
∴ The
![\[center\text{ }is\text{ }\left( 1,\text{ }0 \right),\text{ }eccentricity\text{ }\left( e \right)\text{ }=\text{ }2\surd 3/3,\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-88707cad3eebc91b3cc29dd4eeb82942_l3.png "Rendered by QuickLaTeX.com")
![\[~Foci\text{ }=\text{ }\left( 1~\pm ~2\surd 3,\text{ }0 \right),\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-26c663d8ca53f1da076bf0361149e127_l3.png "Rendered by QuickLaTeX.com")
![\[Equation\text{ }of\text{ }directrix\text{ }=X\text{ }=\text{ }1\pm 9/2\surd 3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6ea548e108a8429257b3411b3c7649a2_l3.png "Rendered by QuickLaTeX.com")