Identify the vertices and the foci.
-$$- 10 x ^ { 2 } + 8 y ^ { 2 } = - 80$$
A) vertices: $( \sqrt { 10 } , 0 ) , ( - \sqrt { 10 } , 0 )$
foci: $( 3 \sqrt { 2 } , 0 ) , ( - 3 \sqrt { 2 } , 0 )$
B) vertices: $( 0 , \sqrt { 10 } ) , ( 0 , - \sqrt { 10 } )$
foci: $( 0,3 \sqrt { 2 } ) , ( 0 , - 3 \sqrt { 2 } )$
C) vertices: $( \sqrt { 8 } , 0 ) , ( - \sqrt { 8 } , 0 )$
foci: $( \sqrt { 2 } , 0 ) , ( - \sqrt { 2 } , 0 )$
D) vertices: $( \sqrt { 8 } , 0 ) , ( - \sqrt { 8 } , 0 )$
foci: $( 3 \sqrt { 2 } , 0 ) , ( - 3 \sqrt { 2 } , 0 )$

Question

Quizplus · Accepted Answer

First, rewrite the equation in standard form for a hyperbola by dividing all terms by -80, resulting in $x^2/8 - y^2/10 = 1$. This indicates a horizontal hyperbola. The vertices are at $(\pm\sqrt{8}, 0)$, and using the formula $c^2 = a^2 + b^2$ (where $c$ is the distance from the center to a focus, $a$ is the distance from the center to a vertex, and $b^2$ is the denominator of the y-term), we find the foci at $(\pm 3\sqrt{2}, 0)$.

Identify the Vertices and the Foci −10x2+8y2=−80- 10 x ^ { 2 } + 8 y ^ { 2 } = - 80−10x2+8y2=−80

Identify the Vertices and the Foci $- 10 x ^ { 2 } + 8 y ^ { 2 } = - 80$