The point (8,2) is on an asyptote of the hyperbola x^2/a^2-y^2/b^2=1. What is |a/b|?

The key to this problem lies in understanding the properties of asymptotes of a hyperbola.

 

Asymptotes and Hyperbolas: A hyperbola has two asymptotes that intersect at the center of the hyperbola. As the hyperbola approaches its branches towards positive or negative infinity, it gets closer and closer to these asymptotes but never touches them.

 

Equation of Asymptotes: The equations of the asymptotes of a hyperbola with a horizontal transverse axis (like the one we're dealing with) can be expressed as:

 

y = ± (x * (b/a)) + k (where k is the y-coordinate of the center)

 

Point on Asymptote: We are given that the point (8, 2) lies on one of the asymptotes. This point will satisfy the equation of the specific asymptote it falls on.

 

Solve for a/b: Since the point (8, 2) is on an asymptote, we can substitute its coordinates (x = 8 and y = 2) into the equation of the asymptote (one of the ± signs will work):

 

2 = ± (8 * (b/a)) + k (We don't need to know the value of k for this problem)

 

Case 1: Positive Sign

 

2 = (8 * (b/a)) + k Assuming k = 0 (since we don't know its value), we have: 2 = 8 * (b/a) (b/a) = 2/8 = 1/4

 

Case 2: Negative Sign

 

2 = -(8 * (b/a)) + k Again, assuming k = 0, we have: 2 = - 8 * (b/a) (b/a) = -2/8 = -1/4 (We negate this value since it's the negative case)

 

Since we only care about the absolute value of a/b, both cases lead to the same answer:

 

|a/b| = 1/4