Find the equation for the ellipse that satisfies the given conditions: Major axis on the $x-$ axis and passes through the points $(4,\,3)$ and $(6,\,2)$
Find the equation for the ellipse that satisfies the given conditions: Major axis on the $x-$ axis and passes through the points $(4,\,3)$ and $(6,\,2)$
since the major axis is on the $x-$ axis, the equation of the ellipse will be of the form
$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ .......... $(1)$
Where, a is the semi-major axis
The ellipse passes through points $(4,\,3)$ and $(6,\,2)$ . Hence,
$\frac{16}{a^{2}}+\frac{9}{b^{2}}=1$ .......... $(2)$
$\frac{36}{a^{2}}+\frac{4}{b^{2}}=1$ .......... $(3)$
On solving equations $(2)$ and $(3),$ we obtain $a^{2}=52$ and $b^{2}=13$
Thus, the equation of the ellipse is $\frac{x^{2}}{52}+\frac{y^{2}}{13}=1$ or $x^{2}+4 y^{2}=52$
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- [JEE MAIN 2020]