Find an equation for the conic that satisfies the given conditions. hyperbola, foci (0, ± $6$ ) , vertices (0, ± $3$ )
A) $6 x ^ { 2 } = y$
B) $\frac { ( x - 6 ) ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 27 } = 1$
C) $\frac { 6 } { 27 } x ^ { 2 } = y$
D) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 27 } = 1$
E) $\frac { y ^ { 2 } } { 36 } - \frac { x ^ { 2 } } { 27 } = 1$

Question

Quizplus · Accepted Answer

The given information tells us that the center of the hyperbola is at (0,0), and the distance from the center to each focus and vertex is 3. This means that the distance from the center to the transverse axis (the line connecting the vertices) is 3 and the distance from the center to the conjugate axis (the line perpendicular to the transverse axis) is 2 (using the Pythagorean theorem). 
Therefore, the equation of the hyperbola is of the form $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$, where $a=3$ and $b=2$. Plugging in these values, we get $\frac{y^2}{36}-\frac{x^2}{27}=1$. Rearranging terms, we get $\frac{y^2}{36}-\frac{x^2}{27}=1$, which matches choice E.

Find an Equation for the Conic That Satisfies the Given 666

Find an Equation for the Conic That Satisfies the Given $6$