Find the foci of the ellipse.
-$$81 x ^ { 2 } + 36 y ^ { 2 } = 2916$$
A) $( 0 , \pm 9 )$
B) $( 0 , \pm 6 )$
C) $( \pm 3 \sqrt { 5 } , 0 )$
D) $( 0 , \pm 3 \sqrt { 5 } )$

Question

Quizplus · Accepted Answer

The standard form of an ellipse centered at the origin is given by $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively. To put our given ellipse in this form, we first divide both sides by 2916 to get $$\frac{x^2}{36} + \frac{y^2}{81} = 1.$$ So $a^2 = 81$ and $b^2 = 36$, and we know that the distance from the center to each focus is given by $c = \sqrt{a^2 - b^2}$. Thus $c = 3\sqrt{5}$, and the foci lie along the major axis, which is parallel to the $x$-axis. Therefore, the foci are at $(0,\pm 3\sqrt{5})$.

Find the Foci of the Ellipse 81x2+36y2=291681 x ^ { 2 } + 36 y ^ { 2 } = 291681x2+36y2=2916

Find the Foci of the Ellipse $81 x ^ { 2 } + 36 y ^ { 2 } = 2916$