To find the standard form of the equation of an ellipse, we need the following information:

1. Center: The center $(h, k)$ is the midpoint of the line segment between the co-vertices or the foci.

2. Semi-major axis: This is the distance from the center to one of the vertices.

3. Semi-minor axis: This is the distance from the center to one of the co-vertices.

4. Equation: The standard form of an ellipse with vertical major axis is: $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$ Where:

   • $a$ is the semi-major axis (vertical direction).
   • $b$ is the semi-minor axis (horizontal direction).
   • $(h,k)$ is the center.

Step 1: Find the center

The co-vertices are $(2,0)$ and $(6,0)$. The center is the midpoint of the co-vertices: $h = \frac{2 + 6}{2} = 4, \quad k = 0$ So, the center of the ellipse is $(4, 0)$.

Step 2: Find the semi-major and semi-minor axes

• Semi-minor axis: The distance from the center $(4,0)$ to one of the co-vertices $(2,0)$ or $(6,0)$ is: $b = 4 - 2 = 2$

• Distance between the center and the foci: The foci are at $(3,0)$ and $(3,4)$. The distance from the center $(4,0)$ to one of the foci $(3,0)$ is: $c = 4 - 3 = 1$

• Relationship between semi-major axis $a$, semi-minor axis $b$, and the distance to the foci $c$ is given by: $c^2 = a^2 - b^2$ Substituting $c = 1$ and $b = 2$: $1^2 = a^2 - 2^2 \quad \Rightarrow \quad a^2 = 5 \quad \Rightarrow \quad a = \sqrt{5}$

Step 3: Write the equation

Since the major axis is vertical, the standard form of the equation of the ellipse is: $\frac{(x - 4)^2}{2^2} + \frac{y^2}{5} = 1$ Simplified, the equation becomes: $\frac{(x - 4)^2}{4} + \frac{y^2}{5} = 1$

This is the standard form of the equation of the ellipse.

Would you like further explanation on any steps?

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Here are five related questions to expand on this:

1. How would the equation change if the major axis was horizontal?
2. How do you find the equation of a hyperbola given its foci and vertices?
3. What is the general form of an ellipse, and how does it relate to the standard form?
4. How do you determine the eccentricity of an ellipse?
5. Can you find the length of the major and minor axes directly from the equation?

Tip: Always check the orientation (horizontal or vertical) of the major axis based on the foci or vertices given, as it changes the standard form of the equation.