The center of a hyperbola is located at the origin. One focus is located at $(-50,0)$ and its associated directrix is represented by the line [tex]x=\frac{2,304}{50}[/tex]. What is the equation of the hyperbola?
Answer (2)
The equation of the hyperbola centered at the origin with a focus at (-50, 0) and a directrix of x = 2304/50 is 2304 x 2 − 196 y 2 = 1 . To derive this, we found a = 48 and b = 14 .
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Determine the value of a using the relationship between the focus and directrix: a = 48 .
Calculate the value of b using the equation c 2 = a 2 + b 2 , where c = 50 , giving b = 14 .
Substitute the values of a 2 and b 2 into the standard equation of a hyperbola with a horizontal transverse axis.
State the equation of the hyperbola: 2304 x 2 − 196 y 2 = 1 .
Explanation
Problem Analysis The problem provides the location of a focus and the equation of the corresponding directrix of a hyperbola centered at the origin. Our goal is to find the equation of this hyperbola. From the given information, we can determine the values of a and b , which define the hyperbola's equation.
Finding a Since the focus is at ( − 50 , 0 ) , we know that c = 50 . The directrix is given by x = 50 2304 , and we know that the equation of the directrix is also given by x = c a 2 . Therefore, we can set up the equation c a 2 = 50 2304 Substituting c = 50 , we get 50 a 2 = 50 2304 Multiplying both sides by 50, we find a 2 = 2304 Taking the square root of both sides, we get a = 2304 = 48
Finding b Now that we have a = 48 and c = 50 , we can use the relationship c 2 = a 2 + b 2 to find b 2 . Substituting the values of a and c , we get 5 0 2 = 4 8 2 + b 2 2500 = 2304 + b 2 Subtracting 2304 from both sides, we find b 2 = 2500 − 2304 = 196 Taking the square root of both sides, we get b = 196 = 14
Equation of the Hyperbola Now we have a = 48 and b = 14 . Since the focus is on the x-axis, the standard equation of the hyperbola is a 2 x 2 − b 2 y 2 = 1 Substituting the values of a 2 and b 2 , we get 4 8 2 x 2 − 1 4 2 y 2 = 1 2304 x 2 − 196 y 2 = 1
Final Answer Therefore, the equation of the hyperbola is 2304 x 2 − 196 y 2 = 1
Examples
Understanding hyperbolas is crucial in various fields. For instance, in physics, the trajectory of a comet as it approaches and recedes from the sun often follows a hyperbolic path. Similarly, in engineering, the design of cooling towers in power plants often involves hyperbolic structures due to their inherent strength and stability. By studying hyperbolas, we gain insights into these real-world phenomena and can apply this knowledge to solve practical problems.
