Finish converting the general form of a hyperbola to standard form in order to identify the type of transverse axis, the center, and the coordinates of the foci.
[tex]
\begin{aligned}
x^2-4 y^2-4 x-48 y-144=0 & \\
x^2-4 x-4 y^2-48 y & =144 \\
\left(x^2-4 x\right)-4\left(y^2+12 y\right) & =144 \\
\left(x^2-4 x+4\right)-4\left(y^2+12 y+36\right) & =144+4-144 \\
(x-2)^2-4(y+6)^2 & =4
\end{aligned}
[/tex]
The transverse axis for this hyperbola is
The center is located at [$\square$]
The coordinates of the foci are
[$\square$]
Answer (1)
Divide the given equation by 4 to obtain the standard form: 4 ( x − 2 ) 2 − 1 ( y + 6 ) 2 = 1 .
Identify the center as ( h , k ) = ( 2 , − 6 ) .
Determine that the transverse axis is horizontal since the x 2 term is positive.
Calculate c = a 2 + b 2 = 4 + 1 = 5 , and find the foci at ( 2 ± 5 , − 6 ) .
Explanation
Analyze the problem We are given the equation of a hyperbola in a partially converted form: ( x − 2 ) 2 − 4 ( y + 6 ) 2 = 4 Our goal is to convert this equation to standard form, identify the transverse axis, the center, and the coordinates of the foci.
Convert to standard form To convert the equation to standard form, we divide both sides by 4: 4 ( x − 2 ) 2 − 4 4 ( y + 6 ) 2 = 4 4 4 ( x − 2 ) 2 − 1 ( y + 6 ) 2 = 1 Now the equation is in the standard form a 2 ( x − h ) 2 − b 2 ( y − k ) 2 = 1 .
Identify the center From the standard form, we can identify the center of the hyperbola as ( h , k ) = ( 2 , − 6 ) .
Determine the transverse axis Since the x 2 term is positive, the transverse axis is horizontal.
Calculate c We have a 2 = 4 and b 2 = 1 , so a = 2 and b = 1 . To find the foci, we need to calculate c , where c 2 = a 2 + b 2 . c 2 = 4 + 1 = 5 c = 5
Find the foci The foci are located at ( h ± c , k ) = ( 2 ± 5 , − 6 ) . Therefore, the coordinates of the foci are ( 2 + 5 , − 6 ) and ( 2 − 5 , − 6 ) .
Examples
Understanding hyperbolas is crucial in various fields, such as physics and engineering. For instance, the trajectory of a comet as it approaches and leaves the sun follows a hyperbolic path. Similarly, the design of cooling towers in nuclear power plants often involves hyperbolic structures for optimal strength and stability. By analyzing the properties of hyperbolas, engineers can accurately predict the behavior of objects moving at high speeds or design structures that can withstand extreme conditions.
