Question

which expression gives the length of the transverse axis of the hyperbola shown below?

a. \\(x + y\\)
b. \\(x - y\\)
c. \\(\frac{x + y}{2}\\)
d. \\(\frac{x - y}{2}\\)

Explanation:

Recall hyperbola definition

A hyperbola is defined geometrically as the set of all points in a plane such that the absolute value of the difference of the distances from two fixed points (the foci) is constant. This constant difference is equal to the length of the transverse axis, \(2a\).

Identify given distances

In the provided diagram, a point on the right branch of the hyperbola is connected to both foci. The distance from this point to the left focus is labeled \(x\), and the distance from this point to the right focus is labeled \(y\).

Relate distances to transverse axis

Using the Geometric Definitions of conic sections, the absolute difference between the focal distances to any point on the hyperbola is equal to the length of the transverse axis:

$$\text{Length of transverse axis} = |x - y|$$

Since the diagram shows \(x > y\), the positive difference is:

$$x - y$$

Answer:

• A. \(x + y\)
• B. \(x - y\) (Correct answer)
• C. \(\frac{x + y}{2}\)
• D. \(\frac{x - y}{2}\)