What are the foci of the graph $25 y^2-4 x^2=100$?
Answer (1)
The equation 25 y 2 − 4 x 2 = 100 represents a hyperbola. By rewriting the equation in standard form 4 y 2 − 25 x 2 = 1 , we identify a 2 = 4 and b 2 = 25 . Since it's a vertical hyperbola, the foci are at ( 0 , ± c ) , where c 2 = a 2 + b 2 = 29 . Thus, c = 29 ≈ 5.4 , and the foci are ( 0 , ± 5.4 ) . The foci of the given hyperbola are ( 0 , ± 5.4 ) .
Explanation
Problem Analysis We are given the equation of a hyperbola 25 y 2 − 4 x 2 = 100 and asked to find the coordinates of its foci.
Rewriting in Standard Form To find the foci, we first need to rewrite the equation in standard form. We divide both sides of the equation by 100: 100 25 y 2 − 100 4 x 2 = 100 100 which simplifies to 4 y 2 − 25 x 2 = 1.
Identifying Parameters This is the standard form of a hyperbola with a vertical transverse axis, where a 2 = 4 and b 2 = 25 . Thus, a = 2 and b = 5 . The foci of a hyperbola with a vertical transverse axis are located at ( 0 , ± c ) , where c 2 = a 2 + b 2 .
Calculating c We calculate c 2 : c 2 = a 2 + b 2 = 4 + 25 = 29. Therefore, c = 29 .
Finding the Foci The foci are at ( 0 , ± 29 ) . Since 29 ≈ 5.385 , the foci are approximately at ( 0 , ± 5.385 ) . Among the given options, ( 0 , ± 5.4 ) is the closest.
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 structural integrity and airflow. By determining the foci of a hyperbola, engineers can accurately model and predict the behavior of these systems, ensuring safety and efficiency.
