What are the foci of the graph $\frac{x^2}{36}-\frac{y^2}{16}=1$?
A. $(0, \pm 7.2)$
B. $( \pm 7.2,0 )$
C. $( \pm 4.5,0 )$
D. $(0, \pm 4.5)$
Answer (1)
Identify the values of a 2 and b 2 from the given equation: a 2 = 36 and b 2 = 16 .
Calculate c 2 using the formula c 2 = a 2 + b 2 , which gives c 2 = 36 + 16 = 52 .
Find c by taking the square root of c 2 : c = 52 = 2 13 ≈ 7.2 .
State the foci of the hyperbola as ( ± c , 0 ) , which are approximately ( ± 7.2 , 0 ) .
Explanation
Problem Analysis We are given the equation of a hyperbola: 36 x 2 − 16 y 2 = 1 . Our goal is to find the foci of this hyperbola.
Identify a and b The standard form of a hyperbola centered at the origin is a 2 x 2 − b 2 y 2 = 1 . In our case, we have a 2 = 36 and b 2 = 16 . Therefore, a = 6 and b = 4 .
Calculate c^2 For a hyperbola, the distance from the center to each focus is denoted by c , and it is related to a and b by the equation c 2 = a 2 + b 2 . Substituting the values of a and b , we get c 2 = 36 + 16 = 52 .
Calculate c Taking the square root of both sides, we find c = 52 = 4 ⋅ 13 = 2 13 .
Determine the foci Since the hyperbola is centered at the origin and has the form a 2 x 2 − b 2 y 2 = 1 , the foci are located at ( ± c , 0 ) . Therefore, the foci are at ( ± 2 13 , 0 ) .
Approximate the value of c We can approximate 2 13 as 2 × 3.60555 ≈ 7.2111 . Thus, the foci are approximately at ( ± 7.2 , 0 ) .
Examples
Understanding hyperbolas and their foci is crucial in various fields, such as astronomy, where the orbits of some comets are hyperbolic. Knowing the foci helps predict the comet's path as it approaches and recedes from the sun. In engineering, the properties of hyperbolas are used in designing lenses and reflectors, where the foci play a key role in focusing light or sound waves. For example, hyperbolic mirrors can focus parallel rays of light to a single point, which is one of the foci.
