Given:
Center is at the origin
and Major axis is along x – axis
So, Equation of ellipse is of the form
![\[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d32e615a6517b76e65b04fd25ae8bf0a_l3.png "Rendered by QuickLaTeX.com")
…(i)
Given that ellipse passing through the points
![\[\left( \mathbf{4},\text{ }\mathbf{3} \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6e14255e5f2d2cbc940379e069bb5844_l3.png "Rendered by QuickLaTeX.com")
and
![\[(-1,4)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-aeaab2e61affb979c28d3883772be7c5_l3.png "Rendered by QuickLaTeX.com")
So, point
and
will satisfy the eq. (i)
Taking point
where x =
![\[4\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-3c0fa656ae249b0b1e03c8cd2e7be583_l3.png "Rendered by QuickLaTeX.com")
and y =
![\[3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-df8bcc09db12acd0f6bda88b4b8135fb_l3.png "Rendered by QuickLaTeX.com")
Putting the values in eq. (i), we get
![\[\frac{{{(4)}^{2}}}{{{a}^{2}}}+\frac{{{(3)}^{2}}}{{{b}^{2}}}=1\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c032a31beda708508259572a55ece321_l3.png "Rendered by QuickLaTeX.com")
![\[\frac{16}{{{a}^{2}}}+\frac{9}{{{b}^{2}}}=1\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-74969bbe6d226c04f667a477bda2a6bf_l3.png "Rendered by QuickLaTeX.com")
…(ii)
Taking point
where x =
![\[-1\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-644e6ba3fd2149684207ec19c2f45db3_l3.png "Rendered by QuickLaTeX.com")
and y =
Putting the values in eq. (i), we get
![\[\frac{16}{{{a}^{2}}}+\frac{16\times 16}{{{b}^{2}}}=16\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d68c2b06ebae544c326136c3ed6391f6_l3.png "Rendered by QuickLaTeX.com")
![\[\frac{16}{{{a}^{2}}}+\frac{256}{{{b}^{2}}}=16\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-da8ce9ee64db653b82814e6f33cd86a5_l3.png "Rendered by QuickLaTeX.com")
…(iii)
Now, we have to solve the above two equations to find the value of a and b Multiply the eq. (iii) by 16, we get
…(iv)
Subtracting eq. (iv) from (ii), we get
![\[<span class="ql-right-eqno"> (1) </span><span class="ql-left-eqno"> </span><img src="https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-2d8d38e21d8df273271b01526fdafeed_l3.png" height="215" width="206" class="ql-img-displayed-equation quicklatex-auto-format" alt="\begin{align*} & \frac{9-256}{{{b}^{2}}}=-15 \\ & -\frac{247}{{{b}^{2}}}=-15 \\ & {{b}^{2}}=\frac{247}{15} \\ \end{align*}" title="Rendered by QuickLaTeX.com"/>\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9705c51839ccc44f424adf9c799dac5f_l3.png "Rendered by QuickLaTeX.com")
Substituting the value of
![\[{{b}^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-22d612841e36c8a3af55a041fef3055b_l3.png "Rendered by QuickLaTeX.com")
in eq. (iii), we get
TThus,
![\[{{a}^{2}}=\frac{247}{7}\,\,\,\And \,\,{{b}^{2}}=\frac{247}{15}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4db85be5b80dce0c085bc139f915e961_l3.png "Rendered by QuickLaTeX.com")
Substituting the value of
![\[{{a}^{2}}and\text{ }{{b}^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-f18222cca635e12bc75738380c9cc066_l3.png "Rendered by QuickLaTeX.com")
in eq. (i), we get
![\[<span class="ql-right-eqno"> (2) </span><span class="ql-left-eqno"> </span><img src="https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-03e3a2b095eebf414e0cb221ae9cd61b_l3.png" height="261" width="570" class="ql-img-displayed-equation quicklatex-auto-format" alt="\begin{align*}</strong> <strong> & \frac{{{x}^{2}}}{\frac{247}{7}}+\frac{{{y}^{2}}}{\frac{247}{15}}=1 \\ </strong> <strong> & \frac{7{{x}^{2}}}{247}+\frac{15{{y}^{2}}}{247}=1 \\ </strong> <strong> & 7{{x}^{2}}+15{{y}^{2}}=247 \\ </strong> <strong>\end{align*}" title="Rendered by QuickLaTeX.com"/>\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-dff57ed556ae0c21f568699c32ad1f28_l3.png "Rendered by QuickLaTeX.com")