Find the equation of the ellipse passing through the point $(3, 5)$ . and length of major axis as $10$

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Solution

Length of major axis is $2 a = 10 a = 5$
The equation of ellipse is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$
It passes through $(3, 5)$
$\frac{3^{2}}{5^{2}} + \frac{5^{2}}{b^{2}} = 1 \frac{9}{25} + \frac{25}{b^{2}} = 1 \frac{25}{b^{2}} = 1 - \frac{9}{25} \frac{25}{b^{2}} = \frac{16}{25} b^{2} = \frac{625}{16} b = \frac{25}{4} ⟹ \frac{x^{2}}{25} + \frac{16 y^{2}}{625} = 1$

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