write the standard form of the equation of ellipse given that F1 (0,0) and F2 (8,0) and has a major axis lenght of 16

To find the standard form of the equation of an ellipse, we need to use the information provided:

1. **Foci**: \( F_1(0, 0) \) and \( F_2(8, 0) \)
2. **Major Axis Length**: 16

### Step 1: Determine the Center and the Distance Between the Foci

The foci of the ellipse are located at \( (0, 0) \) and \( (8, 0) \). The center of the ellipse, which lies on the midpoint of the line segment joining the foci, can be calculated as:

\[
\text{Center} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{0 + 8}{2}, \frac{0 + 0}{2} \right) = (4, 0)
\]

Thus, the center of the ellipse is at \( (4, 0) \).

The distance between the foci \( c \) is:

\[
c = \frac{\text{distance between foci}}{2} = \frac{8 - 0}{2} = 4
\]

### Step 2: Determine the Lengths of the Major and Minor Axes

The length of the major axis is 16, so the semi-major axis \( a \) is:

\[
a = \frac{16}{2} = 8
\]

### Step 3: Calculate the Semi-Minor Axis \( b \)

The relationship between the semi-major axis \( a \), semi-minor axis \( b \), and the distance from the center to each focus \( c \) is given by:

\[
c^2 = a^2 - b^2
\]

Substitute the values of \( c \) and \( a \):

\[
4^2 = 8^2 - b^2
\]

\[
16 = 64 - b^2
\]

\[
b^2 = 64 - 16 = 48
\]

\[
b = \sqrt{48} = 4\sqrt{3}
\]

### Step 4: Write the Standard Form of the Ellipse Equation

Since the major axis is horizontal, the standard form of the equation of the ellipse is:

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

where \( (h, k) \) is the center of the ellipse, \( a = 8 \), and \( b = 4\sqrt{3} \).

Substitute the values:

\[
\frac{(x - 4)^2}{8^2} + \frac{y^2}{(4\sqrt{3})^2} = 1
\]

Simplify the equation:

\[
\frac{(x - 4)^2}{64} + \frac{y^2}{48} = 1
\]

### Final Answer

The standard form of the equation of the ellipse is:

\[
\frac{(x - 4)^2}{64} + \frac{y^2}{48} = 1
\]

Would you like more details or have any questions?

### Related Questions

1. How to convert the standard form of an ellipse to its general form?
2. What are the properties of an ellipse and its applications?
3. How do you find the eccentricity of an ellipse?
4. How to graph an ellipse from its equation?
5. How do the distances between the foci and any point on the ellipse relate?

### Tip

The sum of the distances from any point on an ellipse to its two foci is constant and equal to the length of the major axis.

Write the standard form of the equation of an ellipse given that F1 (0,0) and F2 (8,0) and has a major axis length of 16.

Learn how to derive the standard form of the equation of an ellipse with foci at (0,0) and (8,0) and a major axis length of 16. This problem covers key geometric concepts and formulas, suitable for students in Grades 10-12.

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