Find the equation for the ellipse that satisfies the given conditions:

Question:

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).

Solution:

Since the major axis is on the x-axis, the equation of the ellipse will be of the form

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ $\ldots(1)$

Where, $a$ is the semi-major axis

The ellipse passes through points (4, 3) and (6, 2). Hence,

$\frac{16}{a^{2}}+\frac{9}{b^{2}}=1$ $\ldots(2)$

$\frac{36}{a^{2}}+\frac{4}{b^{2}}=1$ $\ldots(3)$

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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