Problem 1: Equation of the Hyperbola

Given:

• Center: $(0, 0)$
• Focus: $(0, \pm 10)$
• Vertex: $(0, \pm 6)$

Step 1: Hyperbola Orientation and Standard Form

The hyperbola opens vertically because both the focus and vertex are along the $y$-axis. The standard form of such a hyperbola is:
$\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$
Here:

• $a$ is the distance from the center to a vertex.
• $c$ is the distance from the center to a focus.
• The relationship $c^2 = a^2 + b^2$ holds for hyperbolas.

Step 2: Determine $a^2$, $c^2$, and $b^2$

• $a = 6$, so $a^2 = 36$.
• $c = 10$, so $c^2 = 100$.
  Using $c^2 = a^2 + b^2$:
  $100 = 36 + b^2 \implies b^2 = 64.$

Step 3: Equation of the Hyperbola

Substitute $a^2$ and $b^2$ into the standard form:
$\frac{y^2}{36} - \frac{x^2}{64} = 1.$

Problem 2: Equation of the Ellipse

Given:

• Center: $(0, 0)$
• Focus: $(0, \pm 24)$
• Vertex: $(0, \pm 25)$

Step 1: Ellipse Orientation and Standard Form

The ellipse is vertical, so the standard form is:
$\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1$
Here:

• $a$ is the distance from the center to a vertex.
• $c$ is the distance from the center to a focus.
• The relationship $c^2 = a^2 - b^2$ holds for ellipses.

Step 2: Determine $a^2$, $c^2$, and $b^2$

• $a = 25$, so $a^2 = 625$.
• $c = 24$, so $c^2 = 576$.
  Using $c^2 = a^2 - b^2$:
  $576 = 625 - b^2 \implies b^2 = 49.$

Step 3: Equation of the Ellipse

Substitute $a^2$ and $b^2$ into the standard form:
$\frac{y^2}{625} + \frac{x^2}{49} = 1.$

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Final Equations

1. Hyperbola:
   $\frac{y^2}{36} - \frac{x^2}{64} = 1.$

2. Ellipse:
   $\frac{y^2}{625} + \frac{x^2}{49} = 1.$

Would you like further assistance with graphing these equations?

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Related Questions:

1. What is the geometric significance of $a$, $b$, and $c$ in hyperbolas and ellipses?
2. How do you differentiate between the equations of ellipses and hyperbolas?
3. What steps are involved in graphing a hyperbola by hand?
4. How do the foci influence the shape of an ellipse or hyperbola?
5. What are the asymptotes of the hyperbola, and how are they calculated?

Tip: For hyperbolas, always determine the slopes of the asymptotes—they guide accurate graphing!