Is this the equation of a horizontal hyperbola?
$\frac{(x-1)^2}{25}-\frac{(y+3)^2}{9}=1$
A. True
B. False
Answer (1)
The equation is in the form a 2 ( x − h ) 2 − b 2 ( y − k ) 2 = 1 .
Identify the center as ( h , k ) = ( 1 , − 3 ) .
The x term is positive, and the y term is negative, indicating a horizontal hyperbola.
The given equation represents a horizontal hyperbola: True
Explanation
Analyze the equation The given equation is 25 ( x − 1 ) 2 − 9 ( y + 3 ) 2 = 1 . We need to determine if this equation represents a horizontal hyperbola.
Recall the standard form The standard form of a horizontal hyperbola is a 2 ( x − h ) 2 − b 2 ( y − k ) 2 = 1 , where ( h , k ) is the center of the hyperbola.
Compare with the standard form Comparing the given equation 25 ( x − 1 ) 2 − 9 ( y + 3 ) 2 = 1 with the standard form a 2 ( x − h ) 2 − b 2 ( y − k ) 2 = 1 , we can identify the following: h = 1 k = − 3 a 2 = 25 b 2 = 9 The center of the hyperbola is ( 1 , − 3 ) .
Check the signs of the terms In the given equation, the x term is positive and the y term is negative. This indicates that the hyperbola opens horizontally. Therefore, the given equation represents a horizontal hyperbola.
Conclusion The equation 25 ( x − 1 ) 2 − 9 ( y + 3 ) 2 = 1 represents a horizontal hyperbola.
Examples
Understanding hyperbolas is crucial in various fields, such as physics and engineering. For example, 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 shapes to optimize airflow and structural stability. By studying hyperbolas, we can model and analyze these real-world phenomena effectively.
