find the standard equation of foci (2 −√33, 8) and (2 + √33, 8), the sum of the distances of any point from the foci is 14

We are given the foci \( (2 - \sqrt{33}, 8) \) and \( (2 + \sqrt{33}, 8) \), and that the sum of the distances from any point on the ellipse to these two foci is 14.

### Step 1: Recognize the general form of an ellipse
The standard form of an ellipse with horizontal major axis is:
\[
\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
\]
where:
- \( (h, k) \) is the center of the ellipse.
- \( a \) is the semi-major axis (half the total length of the major axis).
- \( b \) is the semi-minor axis.
- The distance between the foci is \( 2c \), where \( c \) is related to \( a \) and \( b \) by the equation \( c^2 = a^2 - b^2 \).

### Step 2: Find the center and distance between the foci
The coordinates of the foci are:
- \( (2 - \sqrt{33}, 8) \)
- \( (2 + \sqrt{33}, 8) \)

The center \( (h, k) \) is the midpoint of the foci:
\[
h = \frac{(2 - \sqrt{33}) + (2 + \sqrt{33})}{2} = 2, \quad k = 8
\]
Thus, the center is \( (2, 8) \).

The distance between the foci, \( 2c \), is:
\[
2c = (2 + \sqrt{33}) - (2 - \sqrt{33}) = 2\sqrt{33}
\]
Therefore, \( c = \sqrt{33} \).

### Step 3: Use the given sum of distances
The sum of the distances from any point on the ellipse to the two foci is equal to \( 2a \). We are given that this sum is 14:
\[
2a = 14 \quad \Rightarrow \quad a = 7
\]

### Step 4: Calculate \( b^2 \)
We know that \( c^2 = a^2 - b^2 \). Since \( a = 7 \) and \( c = \sqrt{33} \), we have:
\[
c^2 = a^2 - b^2 \quad \Rightarrow \quad (\sqrt{33})^2 = 7^2 - b^2
\]
\[
33 = 49 - b^2 \quad \Rightarrow \quad b^2 = 49 - 33 = 16
\]
Thus, \( b = 4 \).

### Step 5: Write the standard equation
Now that we have \( a^2 = 49 \), \( b^2 = 16 \), and the center \( (h, k) = (2, 8) \), the standard equation of the ellipse is:
\[
\frac{(x - 2)^2}{49} + \frac{(y - 8)^2}{16} = 1
\]

This is the standard equation of the ellipse.

---

Would you like further clarification or details? Here are some related questions you might want to explore:

1. How do you find the equation of an ellipse with a vertical major axis?
2. What happens when the sum of distances from the foci changes?
3. How would you find the length of the semi-minor axis from the equation of the ellipse?
4. Can you derive the formula for \( b^2 \) if \( a \), \( c \), and the center are known?
5. How do the foci of a hyperbola differ from those of an ellipse?

**Tip:** The value of \( a \) (the semi-major axis) is always half of the total length of the major axis.

Find the standard equation of foci (2 −√33, 8) and (2 + √33, 8), the sum of the distances of any point from the foci is 14

Find the standard equation of an ellipse with foci at (2 −√33, 8) and (2 + √33, 8) where the sum of distances from any point on the ellipse to the foci is 14. Explore key concepts in ellipse geometry, the distance formula, and the standard equation of ellipses. Suitable for Grades 11-12.

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