Find an equation for the conic that satisfies the given conditions. ellipse, foci $( \pm 1,6 )$, length of major axis 8
A) $6 x ^ { 2 } = y$
B)$\frac { x ^ { 2 } } { 16 } + \frac { ( y - 6 ) ^ { 2 } } { 15 } = 1$
C)$\frac { ( x - 6 ) ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 15 } = 1$
D)$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 15 } = 1$
E) $36 x ^ { 2 } = y$

Question

Quizplus · Accepted Answer

The foci are at $(\pm 1, 6)$, indicating the center of the ellipse is at $(0, 6)$ and the ellipse is oriented along the x-axis. The length of the major axis is 8, so the semi-major axis $a = 4$, giving $a^2 = 16$. The distance from the center to a focus (c) is 1, so $c^2 = 1$. Using the ellipse equation $b^2 = a^2 - c^2$, we find $b^2 = 16 - 1 = 15$. Thus, the equation of the ellipse is $\frac{x^2}{16} + \frac{(y-6)^2}{15} = 1$.

Find an Equation for the Conic That Satisfies the Given (±1,6)( \pm 1,6 )(±1,6)

Find an Equation for the Conic That Satisfies the Given $( \pm 1,6 )$