https://file.aiquickdraw.com/mathbot/file/1727033903382voabr5e3.jpg

Find the standard form of the equation of the ellipse and give the location of its foci.

To analyze the ellipse shown in the image, I will need to extract some key features:

1. **Center**: The ellipse appears to be centered at \((-1, 1)\), which is marked on the graph.

2. **Axes lengths**: Based on the grid, the horizontal axis (major axis) seems to extend 3 units left and right from the center. This means the semi-major axis \(a = 3\). The vertical axis (minor axis) extends 2 units up and down, meaning the semi-minor axis \(b = 2\).

Now we can form the **standard equation of the ellipse** centered at \((h, k)\), with semi-major axis \(a\) and semi-minor axis \(b\):
\[
\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
\]
Substituting the values of the center \((-1, 1)\), \(a = 3\), and \(b = 2\):

\[
\frac{(x + 1)^2}{9} + \frac{(y - 1)^2}{4} = 1
\]

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

### Finding the foci:
The foci are located along the major axis (horizontal axis in this case). The distance from the center to each focus is given by:
\[
c = \sqrt{a^2 - b^2}
\]
Substituting \(a = 3\) and \(b = 2\):
\[
c = \sqrt{9 - 4} = \sqrt{5}
\]
The foci are at \((h \pm c, k)\). Since \(h = -1\) and \(k = 1\), the coordinates of the foci are:
\[
(-1 + \sqrt{5}, 1) \quad \text{and} \quad (-1 - \sqrt{5}, 1)
\]

### Summary:
- **Equation of the ellipse**: \(\frac{(x + 1)^2}{9} + \frac{(y - 1)^2}{4} = 1\)
- **Foci**: \((-1 + \sqrt{5}, 1)\) and \((-1 - \sqrt{5}, 1)\)

Would you like more details or have any questions? Here are 5 related questions for further exploration:

1. How do you determine the orientation of an ellipse (horizontal vs vertical)?
2. What changes in the equation if the major axis is vertical instead of horizontal?
3. How do we find the eccentricity of this ellipse?
4. Can you derive the equation for a hyperbola given a similar diagram?
5. How do we calculate the area of an ellipse?

**Tip**: The foci of an ellipse always lie along the major axis, and the distance between them gives an idea of how "stretched" the ellipse is.

Explore the detailed solution for finding the standard equation of an ellipse centered at (-1, 1) with semi-major axis 3 and semi-minor axis 2. Learn how to calculate the foci location using the ellipse's geometrical properties, suitable for students in Grades 10-12.

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