Find the foci of the hyperbola. $x ^ { 2 } - 5 y ^ { 2 } - 10 = 0$
A) $F ( \pm 5,0 )$
B) $F ( \pm 2 \sqrt { 3 } , 0 )$
C) $F ( 0 , \pm 2 \sqrt { 3 } )$
D) $F ( 0 , \pm \sqrt { 6 } )$
E) $F ( 0 , \pm \sqrt { 3 } )$

Question

Quizplus · Accepted Answer

The standard form of a hyperbola with center at the origin is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, where $a$ and $b$ are the distances from the center to the vertices and from the center to the foci, respectively. By completing the square, we can write $x^2 - 5y^2 - 10 = 0$ as $\frac{x^2}{5} - \frac{y^2}{2} = 1$, from which we see that $a=\sqrt{5}$ and $b=\sqrt{2}$.

Thus, the foci are located at $(\pm c,0)$, where $c=\sqrt{a^2+b^2}$. Applying this formula, we get $c=\sqrt{5+2}=\sqrt{7}$, so the foci are at $(\pm\sqrt{7},0)$.

Therefore, the correct choice is $\boxed{	extbf{(B)}}$.

Find the Foci of the Hyperbola x2−5y2−10=0x ^ { 2 } - 5 y ^ { 2 } - 10 = 0x2−5y2−10=0

Find the Foci of the Hyperbola $x ^ { 2 } - 5 y ^ { 2 } - 10 = 0$