Question

Chose a width for your pool:
The width represents the minor axis of the ellipse. Choose a value for the width, which will correspond to the length of the semi-minor axis. Calculate the value of b
Width 10 feet

B = width/2 = 10/2 = 5 feet

Determine the length:
The length of the pool is twice the width. Use this relationship to calculate the length of the semi-major axis. Calculate the value of a.
Length 16 feet
A = Length/2 = 10/2 = 5 feet

Equation in Standard Form:
Using the values for a and b, write the equation of the ellipse in standard form:
x2a2+y2b2=1

a^2 = 8^2 = 64
b^2 + 5^2 = 25

x^2/64 + y^2/15 = 1

General Equation:
Expand the standard form of the ellipse to write it in general quadratic form:
Ax2+By2+Cx + Dy + E = 0








Foci:
Calculate the coordinates of the foci using the formula: c=a2-b2
Determine the locations of the foci along the major axis (x-axis).

Answer 1

Let's clarify the calculations step by step using the provided dimensions:

### Given:
- **Width (Minor Axis)** = 10 feet
- **Length (Major Axis)** = 16 feet

### Step 1: Calculate semi-minor axis (b)
\[ b = \frac{\text{Width}}{2} = \frac{10}{2} = 5 \text{ feet} \]

### Step 2: Calculate semi-major axis (a)
Since the length of the pool is twice the width, we have:
\[ a = \frac{\text{Length}}{2} = \frac{16}{2} = 8 \text{ feet} \]

### Step 3: Write the equation in standard form
The elliptical equation in standard form is:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
Substituting the values of \( a \) and \( b \):
\[
a^2 = 8^2 = 64 \\
b^2 = 5^2 = 25
\]
Thus, the standard form equation becomes:
\[
\frac{x^2}{64} + \frac{y^2}{25} = 1
\]

### Step 4: Write the general quadratic form
To convert the standard form to general quadratic form, we start with:
\[
\frac{x^2}{64} + \frac{y^2}{25} = 1
\]
Multiply through by \( 64 \times 25 \) (which simplifies the fractions):
\[
25x^2 + 64y^2 = 64 \times 25
\]

Now calculate \( 64 \times 25 = 1600 \) to get:
\[
25x^2 + 64y^2 - 1600 = 0
\]
This is of the form \( Ax^2 + By^2 + Cx + Dy + E = 0 \):
\[
25x^2 + 64y^2 + 0x + 0y - 1600 = 0
\]

### Step 5: Calculate coordinates of the foci
The distance from the center to each focus (c) is calculated using:
\[
c = \sqrt{a^2 - b^2}
\]
\[
c = \sqrt{64 - 25} = \sqrt{39}
\]
Calculating further:
\[
c \approx 6.244 \text{ feet}
\]

### Foci locations along the x-axis
Since this ellipse is oriented along the x-axis, the foci will be located at:
\[
(\pm c, 0) = (\pm \sqrt{39}, 0) \approx (\pm 6.244, 0)
\]

### Summary
1. **Semi-minor axis (b)**: 5 feet
2. **Semi-major axis (a)**: 8 feet
3. **Standard form:** \(\frac{x^2}{64} + \frac{y^2}{25} = 1\)
4. **General Equation:** \(25x^2 + 64y^2 - 1600 = 0\)
5. **Coordinates of the foci:** Approximately \( (6.244, 0) \) and \( (-6.244, 0) \) along the x-axis.