Given: Eccentricity is
![\[\sqrt{2}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5f2888138764fe9f08b2aade2cd23a71_l3.png "Rendered by QuickLaTeX.com")
, and the distance between foci is
![\[16\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a9afcd935e0b07e58e19bb7c3d7d4534_l3.png "Rendered by QuickLaTeX.com")
Need to find: The equation of the hyperbola.
Let, the equation of the hyperbola be:
![\[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d34b8d9c36294615215fbaa34b025265_l3.png "Rendered by QuickLaTeX.com")
Distance between the foci is
, i.e.,
![\[2ae=16\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-90eeb6b30f0add856f0f1a0e4979dd01_l3.png "Rendered by QuickLaTeX.com")
And also given, the eccentricity,
![\[e=\sqrt{2}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6532339ea9229f378a4d77ae6168af0e_l3.png "Rendered by QuickLaTeX.com")
Therefore,
![\[<span class="ql-right-eqno"> (1) </span><span class="ql-left-eqno"> </span><img src="https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5e295f600a2192ad7e18b3bd848b89b3_l3.png" height="87" width="147" class="ql-img-displayed-equation quicklatex-auto-format" alt="\begin{align*} & 2a\sqrt{2}=16 \\ & a=4\sqrt{2} \\ \end{align*}" title="Rendered by QuickLaTeX.com"/>\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-765c38a3f3dd9c3521f52d44c11c7e80_l3.png "Rendered by QuickLaTeX.com")
We know that,
![\[<span class="ql-right-eqno"> (2) </span><span class="ql-left-eqno"> </span><img src="https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9e3bd551222adcc30dbc7e1d3c5b9382_l3.png" height="418" width="208" class="ql-img-displayed-equation quicklatex-auto-format" alt="\begin{align*} & e=\sqrt{1+\frac{{{b}^{2}}}{{{a}^{2}}}} \\ & Therefore, \\ & \sqrt{1+\frac{{{b}^{2}}}{{{a}^{2}}}}=\sqrt{2} \\ & 1+\frac{{{b}^{2}}}{{{a}^{2}}}=2 \\ & \frac{{{b}^{2}}}{{{a}^{2}}}=1 \\ & {{b}^{2}}={{a}^{2}}=32 \\ \end{align*}" title="Rendered by QuickLaTeX.com"/>\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-bf4be319889ef2a4eb376971170b5438_l3.png "Rendered by QuickLaTeX.com")
So, the equation of the hyperbola is
![\[<span class="ql-right-eqno"> (3) </span><span class="ql-left-eqno"> </span><img src="https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7b267cdf3972be54da56b0609250aff7_l3.png" height="247" width="505" class="ql-img-displayed-equation quicklatex-auto-format" alt="\begin{align*}</strong> <strong> & \frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1 \\ </strong> <strong> & \frac{{{x}^{2}}}{32}-\frac{{{y}^{2}}}{32}=1 \\ </strong> <strong> & {{x}^{2}}-{{y}^{2}}=32 \\ </strong> <strong>\end{align*}" title="Rendered by QuickLaTeX.com"/>\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-f30c0fa162653c82c9bd27191a028cdd_l3.png "Rendered by QuickLaTeX.com")