Standard equation of center (-3,-7) major axis length is 20 minor axis is 12

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

1. The center of the ellipse \((h, k)\).
2. The length of the major axis.
3. The length of the minor axis.

From the problem statement, we have:

- Center \((h, k) = (-3, -7)\)
- Length of the major axis = 20
- Length of the minor axis = 12

### Step-by-Step Solution:

1. **Determine the semi-major and semi-minor axes:**

   - Semi-major axis length \(a\) is half the length of the major axis:
     \[
     a = \frac{20}{2} = 10
     \]

   - Semi-minor axis length \(b\) is half the length of the minor axis:
     \[
     b = \frac{12}{2} = 6
     \]

2. **Standard equation of an ellipse:**

   The standard form of an ellipse with center \((h, k)\) is given by:

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

   Plugging in the values:

   - \(h = -3\)
   - \(k = -7\)
   - \(a = 10\), hence \(a^2 = 100\)
   - \(b = 6\), hence \(b^2 = 36\)

   The equation becomes:

   \[
   \frac{(x+3)^2}{100} + \frac{(y+7)^2}{36} = 1
   \]

This is the standard equation of the ellipse with the given center and axis lengths.

### Do you want more details or have any questions?

### Related Questions:

1. How would the equation change if the lengths of the major and minor axes were swapped?
2. What would the equation look like if the center were at the origin?
3. How do you find the foci of the ellipse given the equation?
4. How does the orientation of the major and minor axes affect the equation of an ellipse?
5. How would you graph the ellipse based on the given equation?

### Tip:
Always remember that the semi-major axis \(a\) is associated with the larger of the two axis lengths, and the semi-minor axis \(b\) is associated with the smaller. This helps in correctly identifying \(a^2\) and \(b^2\) in the standard form of the ellipse equation.

Discover how to find the standard equation of an ellipse with center (-3, -7), major axis length 20, and minor axis length 12. Step-by-step solution provided, including detailed calculations and formula application. Suitable for advanced high school students and beyond.

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