foci (2-√33,8) and (2 + √33, 8), the sum of the distances of any point at the foci is 14

The given problem describes the foci of an ellipse and specifies that the sum of the distances from any point on the ellipse to the foci is 14.

The standard equation of an ellipse is given by:

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

Where:
- \((h, k)\) is the center of the ellipse,
- \(2a\) is the length of the major axis,
- \(2b\) is the length of the minor axis,
- The distance between the foci is \(2c\), where \(c^2 = a^2 - b^2\).

### Step 1: Identify the center of the ellipse
The foci are given as \((2 - \sqrt{33}, 8)\) and \((2 + \sqrt{33}, 8)\). Since the foci are symmetric around the center, the center \((h, k)\) of the ellipse is the midpoint of these two points:

\[
h = \frac{(2 - \sqrt{33}) + (2 + \sqrt{33})}{2} = \frac{4}{2} = 2
\]
\[
k = 8
\]

So, the center of the ellipse is \((2, 8)\).

### Step 2: Determine the distance between the foci
The distance between the two foci is given by:

\[
2c = \sqrt{(2 + \sqrt{33} - (2 - \sqrt{33}))^2 + (8 - 8)^2} = \sqrt{(2\sqrt{33})^2} = 2\sqrt{33}
\]

Thus, \(c = \sqrt{33}\).

### Step 3: Use the given information about the major axis
The problem states that the sum of the distances from any point on the ellipse to the foci is 14. This sum is always equal to \(2a\) for an ellipse, so:

\[
2a = 14 \implies a = 7
\]

### Step 4: Find \(b^2\)
Now, we can find \(b^2\) using the relation \(c^2 = a^2 - b^2\):

\[
c^2 = a^2 - b^2
\]
\[
(\sqrt{33})^2 = 7^2 - b^2
\]
\[
33 = 49 - b^2
\]
\[
b^2 = 49 - 33 = 16
\]

### Step 5: Write the equation of the ellipse
Now that we have \(a^2 = 49\), \(b^2 = 16\), and the center \((h, k) = (2, 8)\), the equation of the ellipse is:

\[
\frac{(x - 2)^2}{49} + \frac{(y - 8)^2}{16} = 1
\]

This is the equation of the ellipse.

Would you like more details or have any further questions?

Here are 5 related questions for further understanding:
1. How do you find the equation of a hyperbola given its foci and vertices?
2. What is the difference between an ellipse and a circle in terms of their equations?
3. How can you determine the eccentricity of an ellipse?
4. What is the significance of the major and minor axes in an ellipse?
5. How would the equation of the ellipse change if the foci were aligned vertically?

**Tip**: In an ellipse, the eccentricity \(e = \frac{c}{a}\) is always less than 1, and it indicates how "stretched" the ellipse is.

Learn how to find the equation of an ellipse with foci at (2-√33, 8) and (2 + √33, 8), and a total distance of 14 for any point on the ellipse. This problem explores key concepts in coordinate geometry, conic sections, and the properties of ellipses. Suitable for students in Grades 10-12.

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