Find the equation of the ellipse,whose length | vedclass

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(A) Since the foci are on the $y$-axis,the major axis is along the $y$-axis. The equation of the ellipse is of the form $\frac{x^{2}}{b^{2}} + \frac{y

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$\frac{x^{2}}{75} + \frac{y^{2}}{100} = 1$

Vedclass · Answer

(A) Since the foci are on the $y$-axis,the major axis is along the $y$-axis. The equation of the ellipse is of the form $\frac{x^{2}}{b^{2}} + \frac{y^{2}}{a^{2}} = 1$.Given that the length of the major axis is $2a = 20$,so $a = 10$.The foci are $(0, \pm c) = (0, \pm 5)$,so $c = 5$.Using the relation $c^{2} = a^{2} - b^{2}$,we have $5^{2} = 10^{2} - b^{2}$.$25 = 100 - b^{2}$,which implies $b^{2} = 100 - 25 = 75$.Substituting $a^{2} = 100$ and $b^{2} = 75$ into the standard equation,we get $\frac{x^{2}}{75} + \frac{y^{2}}{100} = 1$.

Find the equation of the ellipse,whose length of the major axis is $20$ and foci are $(0, \pm 5)$.

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