To solve this problem, we need to find the general form of the equation of an ellipse given the following information:

• Center of the ellipse: $(h, k) = (3, -1)$
• $a = 4$ (the semi-major axis)
• $b = 3$ (the semi-minor axis)
• The major axis is horizontal.

General Equation of an Ellipse with a Horizontal Major Axis

The standard form of the equation of an ellipse centered at $(h, k)$ with a horizontal major axis is:

$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$

Where:

• $h = 3$
• $k = -1$
• $a = 4$, so $a^2 = 16$
• $b = 3$, so $b^2 = 9$

Plugging in the values:

$\frac{(x - 3)^2}{16} + \frac{(y + 1)^2}{9} = 1$

This is the general form of the ellipse based on the given conditions.

Would you like a more detailed explanation, or do you have any further questions?

Here are five related questions for further exploration:

1. What changes in the equation if the major axis is vertical instead of horizontal?
2. How would the equation change if the ellipse were shifted to the origin?
3. What is the general form of the equation for a circle, and how does it relate to an ellipse?
4. How can you find the foci of the ellipse given $a$ and $b$?
5. What is the significance of the values of $a$ and $b$ in the context of the ellipse's shape?

Tip: Remember, if the major axis is vertical, the roles of $a^2$ and $b^2$ in the equation of the ellipse are swapped!