Find the center, vertices and foci of the hyperbola. $26 y ^ { 2 } - 23 x ^ { 2 } = 598$
A)Center: (0, 0)Vertices: $( 0 , \pm \sqrt { 7 } )$ Foci: $( 0 , \pm \sqrt { 49 } )$
B)Center: $( 23,26 )$ Vertices: $( 0 , \pm \sqrt { 7 } )$ Foci: $( 0 , \pm \sqrt { 49 } )$
C)Center: (0, 0)Vertices: $( \pm \sqrt { 49 } , 0 )$ Foci: $( 0 , \pm \sqrt { 49 } )$
D)Center: (0, 0)Vertices: $( \pm \sqrt { 7 } , 0 )$ Foci: $( \pm \sqrt { 49 } , 0 )$
E)Center: $( 26,23 )$ Vertices: $( 0 , \pm \sqrt { 7 } )$ Foci: $( 0 , \pm \sqrt { 49 } )$

Question

Find the center, vertices and foci of the hyperbola.  $26 y ^ { 2 } - 23 x ^ { 2 } = 598$ 
A)Center: (0, 0)Vertices: $( 0 , \pm \sqrt { 7 } )$ Foci: $( 0 , \pm \sqrt { 49 } )$
B)Center: $( 23,26 )$  Vertices: $( 0 , \pm \sqrt { 7 } )$ Foci: $( 0 , \pm \sqrt { 49 } )$
C)Center: (0, 0)Vertices: $( \pm \sqrt { 49 } , 0 )$ Foci: $( 0 , \pm \sqrt { 49 } )$
D)Center: (0, 0)Vertices: $( \pm \sqrt { 7 } , 0 )$ Foci: $( \pm \sqrt { 49 } , 0 )$
E)Center: $( 26,23 )$ Vertices: $( 0 , \pm \sqrt { 7 } )$ Foci: $( 0 , \pm \sqrt { 49 } )$

Quizplus · Accepted Answer

The given equation $26y^2 - 23x^2 = 598$ can be rewritten in the standard form of a hyperbola by dividing each term by 598, resulting in $\frac{y^2}{\frac{598}{26}} - \frac{x^2}{\frac{598}{23}} = 1$. This simplifies to $\frac{y^2}{23} - \frac{x^2}{26} = 1$, indicating that the center of the hyperbola is at the origin (0, 0). The vertices are determined by the value under the $y^2$ term, $\sqrt{23}$, along the y-axis, and the foci are found by calculating $c = \sqrt{a^2 + b^2}$, where $a^2 = 23$ and $b^2 = 26$, leading to $c = \sqrt{23 + 26} = \sqrt{49}$, which are located along the y-axis as well. However, there was a mistake in my calculation of the vertices; they should be based directly on the value under the $y^2$ term, which means the vertices are at $(0, \pm \sqrt{23})$, not $(0, \pm \sqrt{7})$, and the foci calculation was correct but misinterpreted in the context of the choices provided. The correct interpretation of the standard form and the components of the hyperbola was intended, but the specific details provided in the answer choices led to an incorrect match.

Find the Center, Vertices and Foci of the Hyperbola 26y2−23x2=59826 y ^ { 2 } - 23 x ^ { 2 } = 59826y2−23x2=598

Find the Center, Vertices and Foci of the Hyperbola $26 y ^ { 2 } - 23 x ^ { 2 } = 598$