To solve this problem, let’s analyze the hyperbola and its equation:

The general equation for a hyperbola in the form shown is:

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

Here:

• $a$ represents the distance from the center of the hyperbola to the vertices along the x-axis.
• $b$ is used to determine the asymptotes, which are given by the equations: $y = \pm \frac{b}{a}x$

Steps:

1. Identify $a$:

   • The vertices are located at the x-intercepts of the hyperbola.
   • Observing the graph, the vertices are at $x = \pm 5$.
   • Therefore, $a = 5$.

2. Identify $b$:

   • The asymptotes are given as dashed lines, and their slopes can be calculated as $\pm \frac{b}{a}$.
   • From the graph, the slope of the asymptotes is $\pm 2$, so: $\frac{b}{a} = 2 \quad \Rightarrow \quad b = 2a = 2 \cdot 5 = 10$

Solution:

$a = 5, \quad b = 10$

Let me know if you'd like further explanation or have any other questions! Here are 5 related questions to explore:

1. How do you derive the equation of the asymptotes of a hyperbola?
2. What is the difference between hyperbolas with horizontal vs. vertical orientations?
3. Can you identify the foci of this hyperbola? (Hint: Use $c^2 = a^2 + b^2$.)
4. How do the equations change for a hyperbola centered at $(h, k)$?
5. What are the real-world applications of hyperbolas in physics or engineering?

Tip: Always double-check the slopes of the asymptotes to confirm the relationship between $a$ and $b$.