Let an ellipse frac{x^2}{a^2} + frac{y^2}{b^2 | vedclass

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(D) Given the ellipse equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a < b$. Since $a < b$,the eccentricity $e$ is given by $e^2 = 1 - \frac{a

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$\frac{8\sqrt{5}}{3}$

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(D) Given the ellipse equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a Since $a Given $e = \frac{\sqrt{5}}{3}$,so $e^2 = \frac{5}{9}$. Thus,$1 - \frac{a^2}{b^2} = \frac{5}{9} \implies \frac{a^2}{b^2} = 1 - \frac{5}{9} = \frac{4}{9}$. Let $a^2 = 4k$ and $b^2 = 9k$ for some constant $k > 0$. The ellipse passes through $(4, 3)$,so $\frac{4^2}{a^2} + \frac{3^2}{b^2} = 1$. Substituting the values,$\frac{16}{4k} + \frac{9}{9k} = 1 \implies \frac{4}{k} + \frac{1}{k} = 1 \implies \frac{5}{k} = 1 \implies k = 5$. Therefore,$a^2 = 4(5) = 20$ and $b^2 = 9(5) = 45$. This gives $a = \sqrt{20} = 2\sqrt{5}$ and $b = \sqrt{45} = 3\sqrt{5}$. The length of the latus rectum for an ellipse with $a Length $= \frac{2(20)}{3\sqrt{5}} = \frac{40}{3\sqrt{5}} = \frac{40\sqrt{5}}{3 	imes 5} = \frac{8\sqrt{5}}{3}$.

Let an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a < b$,pass through the point $(4, 3)$ and have eccentricity $\frac{\sqrt{5}}{3}$. Then the length of its latus rectum is:

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