Write the standard form of the equation of the hyperbola subject to the given conditions.
-Corners of the reference rectangle: $( 2,10 ) , ( 2,0 ) , ( - 10,10 ) , ( - 10,0 )$; Horizontal transverse axis
A) $\frac { ( x - 4 ) ^ { 2 } } { 36 } - \frac { ( y + 5 ) ^ { 2 } } { 25 } = 1$
B) $\frac { ( x + 4 ) ^ { 2 } } { 36 } - \frac { ( y - 5 ) ^ { 2 } } { 25 } = 1$
C) $\frac { ( y - 5 ) ^ { 2 } } { 25 } - \frac { ( x + 4 ) ^ { 2 } } { 36 } = 1$
D) $\frac { ( y + 4 ) ^ { 2 } } { 36 } - \frac { ( x - 5 ) ^ { 2 } } { 25 } = 1$

Question

Quizplus · Accepted Answer

The center of the hyperbola is the midpoint of the rectangle's diagonals, which is at $(-4, 5)$. Since the transverse axis is horizontal, the $x$-term comes first and is positive. The lengths of the semi-major and semi-minor axes are half the lengths of the rectangle's sides, which are $6$ and $5$ respectively, giving $a^2 = 36$ and $b^2 = 25$. Therefore, the correct equation is $\frac{(x + 4)^2}{36} - \frac{(y - 5)^2}{25} = 1$.

Write the Standard Form of the Equation of the Hyperbola (2,10),(2,0),(−10,10),(−10,0)( 2,10 ) , ( 2,0 ) , ( - 10,10 ) , ( - 10,0 )(2,10),(2,0),(−10,10),(−10,0)

Write the Standard Form of the Equation of the Hyperbola $( 2,10 ) , ( 2,0 ) , ( - 10,10 ) , ( - 10,0 )$