Let's solve this systematically:

Part 1: Horizontal Hyperbola

General Equation:

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

• Transverse axis: Horizontal
• Vertices: $(\pm a, 0)$
• Foci: $(\pm c, 0)$, where $c = \sqrt{a^2 + b^2}$

For the specific equation: $\frac{x^2}{42} - \frac{y^2}{32} = 1$

• $a^2 = 42 \implies a = \sqrt{42}$
• $b^2 = 32$
• $c = \sqrt{a^2 + b^2} = \sqrt{42 + 32} = \sqrt{74}$

Solution:

• Vertices: $(\sqrt{42}, 0)$ and $(- \sqrt{42}, 0)$
• Foci: $(\sqrt{74}, 0)$ and $(- \sqrt{74}, 0)$

Part 2: Vertical Hyperbola

General Equation:

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

• Transverse axis: Vertical
• Vertices: $(0, \pm a)$
• Foci: $(0, \pm c)$, where $c = \sqrt{a^2 + b^2}$

For the specific equation: $\frac{y^2}{42} - \frac{x^2}{32} = 1$

• $a^2 = 42 \implies a = \sqrt{42}$
• $b^2 = 32$
• $c = \sqrt{a^2 + b^2} = \sqrt{42 + 32} = \sqrt{74}$

Solution:

• Vertices: $(0, \sqrt{42})$ and $(0, -\sqrt{42})$
• Foci: $(0, \sqrt{74})$ and $(0, -\sqrt{74})$

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Final Answers:

1. For the horizontal hyperbola:

   • Vertices: $(x, y) = (\sqrt{42}, 0)$ and $(x, y) = (-\sqrt{42}, 0)$
   • Foci: $(x, y) = (\sqrt{74}, 0)$ and $(x, y) = (-\sqrt{74}, 0)$

2. For the vertical hyperbola:

   • Vertices: $(x, y) = (0, \sqrt{42})$ and $(x, y) = (0, -\sqrt{42})$
   • Foci: $(x, y) = (0, \sqrt{74})$ and $(x, y) = (0, -\sqrt{74})$

Let me know if you'd like further clarifications or computations!

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Here are related questions for deeper understanding:

1. What distinguishes hyperbolas with horizontal versus vertical transverse axes?
2. How is the relationship between $a$, $b$, and $c$ derived geometrically?
3. What happens when $a = b$ in a hyperbola equation?
4. How can the asymptotes of a hyperbola be determined from its equation?
5. Can the hyperbola equation be rewritten in parametric form?

Tip: Always visualize the hyperbola using its asymptotes and key points (vertices and foci) to understand its shape better.