The ellipse passes through
![\[\left( 1,\text{ }4 \right)\text{ }and\text{ }\left( -\text{ }6,\text{ }1 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-182cec44a2f5816534decde41dca6ea4_l3.png "Rendered by QuickLaTeX.com")
Given:
The points
Now let us find the equation to the ellipse.
We know that the equation of the ellipse whose axes are x and y – axis is given as
![\[\ldots .\text{ }\left( 1 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7725ea4c54cf7a25c9cbb8d6c6a9b13b_l3.png "Rendered by QuickLaTeX.com")
Let us substitute the point (1, 4) in equation (1), we get
![\[{{b}^{2}}~+\text{ }16{{a}^{2}}~=\text{ }{{a}^{2}}~{{b}^{2}}~\ldots .\text{ }\left( 2 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-441f410b85324056ea0ce2adb8680e5f_l3.png "Rendered by QuickLaTeX.com")
Let us substitute the point (-6, 1) in equation (1), we get
![\[{{a}^{2}}~+\text{ }36{{b}^{2}}~=\text{ }{{a}^{2}}{{b}^{2}}~\ldots .\text{ }\left( 3 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-272257696141d870f74b8595abc0177c_l3.png "Rendered by QuickLaTeX.com")
Let us multiply equation (3) by 16 and subtract with equation (2), we get
![\[(16{{a}^{2}}~+\text{ }576{{b}^{2}})\text{ }\text{ }({{b}^{2}}~+\text{ }16{{a}^{2}})\text{ }=\text{ }(16{{a}^{2}}{{b}^{2}}-\text{ }{{a}^{2}}{{b}^{2}})\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-3631e2376c9ef55d4d453ff03dce4d97_l3.png "Rendered by QuickLaTeX.com")
![\[575{{b}^{2}}~=\text{ }15{{a}^{2}}{{b}^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-49dd4a0bbdf1513f156c4f4313a1c3fd_l3.png "Rendered by QuickLaTeX.com")
Or,
![\[15{{a}^{2}}~=\text{ }575\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a25207bffc9ac41012c925685643e6f6_l3.png "Rendered by QuickLaTeX.com")
![\[{{a}^{2}}~=\text{ }575/15\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-cc72a41c7e5f9767aaed2e4af60f1d55_l3.png "Rendered by QuickLaTeX.com")
So,
![\[=\text{ }115/3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-397e1a6ad9115c98885fa4d1ff4588d9_l3.png "Rendered by QuickLaTeX.com")
So from equation (2),
So the equation of the ellipse can be given as
![\[3{{x}^{2}}~+\text{ }7{{y}^{2}}~=\text{ }115\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6484915f9ffc5e21a8314bd1f9b64bac_l3.png "Rendered by QuickLaTeX.com")
∴ The equation of the ellipse is