Can you solve an ellipse and graph it? The given is: Co-vertice (7,-3) and (-1,-3), vertex at (3,3)

Yes, I can help you solve this ellipse and graph it.

### Step 1: Identify the Center of the Ellipse
The co-vertices are given as \((7, -3)\) and \((-1, -3)\). Since co-vertices lie along the minor axis, the center of the ellipse is the midpoint of the line segment connecting the co-vertices. 

\[
\text{Center} = \left( \frac{7 + (-1)}{2}, \frac{-3 + (-3)}{2} \right) = \left( \frac{6}{2}, \frac{-6}{2} \right) = (3, -3)
\]

Thus, the center of the ellipse is \((3, -3)\).

### Step 2: Find the Length of the Minor Axis
The distance between the co-vertices is the length of the minor axis, which is the distance between \((7, -3)\) and \((-1, -3)\). The formula for distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting the co-vertices:

\[
\text{Length of Minor Axis} = \sqrt{(7 - (-1))^2 + (-3 - (-3))^2} = \sqrt{(8)^2 + 0^2} = 8
\]

Thus, the total length of the minor axis is 8, so the semi-minor axis (\(b\)) is half of that:

\[
b = \frac{8}{2} = 4
\]

### Step 3: Find the Length of the Major Axis
The vertices are given as \((3, 3)\) and \((3, -9)\), with the center at \((3, -3)\). The distance between a vertex and the center is the semi-major axis length (\(a\)).

\[
\text{Length of Major Axis} = |3 - (-9)| = 12
\]

Thus, the semi-major axis (\(a\)) is half of this:

\[
a = \frac{12}{2} = 6
\]

### Step 4: Equation of the Ellipse
Since the major axis is vertical and the center is at \((3, -3)\), the equation of the ellipse is of the form:

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

Substituting \(h = 3\), \(k = -3\), \(a = 6\), and \(b = 4\):

\[
\frac{(x - 3)^2}{4^2} + \frac{(y + 3)^2}{6^2} = 1
\]

Simplifying:

\[
\frac{(x - 3)^2}{16} + \frac{(y + 3)^2}{36} = 1
\]

This is the equation of the ellipse.

### Step 5: Graph the Ellipse
Now that we have the equation, we can graph the ellipse. I'll create a graph based on this equation.

Let me generate the graph for you.It seems there was a mistake in plotting the vertices. Let me fix it and regenerate the graph for you.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, we have already solved the equation of the ellipse as:

\[
\frac{(x - 3)^2}{16} + \frac{(y + 3)^2}{36} = 1
\]

Feel free to graph it based on this equation using any graphing tool, or let me know if you'd like any further clarification.

Here are some related questions you might find helpful:
1. How do you determine the equation of an ellipse with different axes lengths?
2. What is the significance of vertices and co-vertices in an ellipse?
3. How do you graph ellipses with horizontal and vertical major axes?
4. How would the equation of the ellipse change if the center were at the origin?
5. Can ellipses be represented in polar coordinates?

**Tip:** When graphing ellipses, always ensure the major and minor axes are correctly scaled for a more accurate plot!

Learn how to solve an ellipse problem with given co-vertices at (7,-3) and (-1,-3) and a vertex at (3,3). This step-by-step solution covers key concepts in conic sections, such as finding the center, semi-major and semi-minor axes, and deriving the equation of the ellipse. Suitable for students in Grades 10-12.

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