Find the vertices and locate the foci for the hyperbola whose equation is given.
-$$16 x ^ { 2 } - 4 y ^ { 2 } = 64$$
A) vertices: $( - 2,0 ) , ( 2,0 )$
foci: $( - 2 \sqrt { 5 } , 0 ) , ( 2 \sqrt { 5 } , 0 )$
B) vertices: $( 0 , - 2 ) , ( 0,2 )$
foci: $( 0 , - 2 \sqrt { 5 } ) , ( 0,2 \sqrt { 5 } )$
C) vertices: $( - 2,0 ) , ( 2,0 )$
foci: $( - 2 \sqrt { 3 } , 0 ) , ( 2 \sqrt { 3 } , 0 )$
D) vertices: $( - 4,0 ) , ( 4,0 )$
foci: $( - 2 \sqrt { 5 } , 0 ) , ( 2 \sqrt { 5 } , 0 )$

Question

Quizplus · Accepted Answer

First, we need to get the equation into standard form:
$$\frac{x^2}{4} - \frac{y^2}{16} = 1$$
From this, we can see that $a=2$ (which gives us the vertices) and $b=4$.
To find the foci, we can use the formula $c^2 = a^2 + b^2$. Plugging in our values gives us $c^2 = 20$, so $c=\sqrt{20}=2\sqrt{5}$. Since the foci are located along the x-axis, their coordinates will be $(\pm c, 0) = (-2\sqrt{5}, 0)$ and $(2\sqrt{5}, 0)$. Therefore, the correct choice is A.

Find the Vertices and Locate the Foci for the Hyperbola 16x2−4y2=6416 x ^ { 2 } - 4 y ^ { 2 } = 6416x2−4y2=64

Find the Vertices and Locate the Foci for the Hyperbola $16 x ^ { 2 } - 4 y ^ { 2 } = 64$