The equation $\frac{y^2}{8^2}-\frac{x^2}{b^2}$ represents a hyperbola centered at the origin with a focus of $(0,-10)$. What is the value of $b$?
A. 4
B. 6
C. 10
D. 12
Answer (2)
The value of b in the hyperbola equation is 6, determined using the relationship c 2 = a 2 + b 2 . Since a = 8 and c = 10 , solving the equation yields b = 6 . The final answer is option B: 6.
;
The given hyperbola equation is 8 2 y 2 − b 2 x 2 = 1 , with a focus at ( 0 , − 10 ) .
Use the relationship c 2 = a 2 + b 2 , where a = 8 and c = 10 .
Substitute the values to get 1 0 2 = 8 2 + b 2 , which simplifies to 100 = 64 + b 2 .
Solve for b , resulting in b = 36 = 6 . The final answer is 6 .
Explanation
Problem Analysis We are given the equation of a hyperbola 8 2 y 2 − b 2 x 2 = 1 centered at the origin with a focus at ( 0 , − 10 ) . We need to find the value of b .
Hyperbola Properties Since the hyperbola is centered at the origin and has a vertical transverse axis, its equation is of the form a 2 y 2 − b 2 x 2 = 1 , where a = 8 . The foci are at ( 0 , ± c ) , where c 2 = a 2 + b 2 .
Finding the relationship between a, b and c We are given that one focus is at ( 0 , − 10 ) , so c = 10 . We can now use the relationship c 2 = a 2 + b 2 to find b .
Substituting values Substitute the values of a and c into the equation: 1 0 2 = 8 2 + b 2 .
Solving for b^2 Solve for b 2 : b 2 = 1 0 2 − 8 2 = 100 − 64 = 36 .
Solving for b Solve for b : b = 36 = 6 .
Final Answer Therefore, the value of b is 6.
Examples
Understanding hyperbolas is crucial in various fields, such as physics (describing the paths of particles in certain force fields) and astronomy (modeling the trajectories of comets). In architecture, hyperbolic structures offer unique strength and aesthetic appeal, allowing for innovative designs in buildings and bridges. By grasping the properties of hyperbolas, students can appreciate their practical applications in both natural phenomena and human-engineered creations.
