To find the equation of the ellipse with the given parameters, we will follow these steps:
### Step 1: Identify the given information
- Eccentricity (e) = 2/3
- Focus (S) = (3, 0)
- Vertex (V) = (1, 0)
### Step 2: Calculate the distance between the focus and the vertex
Using the distance formula, we find the distance \( SV \):
\[
SV = \sqrt{(3 - 1)^2 + (0 - 0)^2} = \sqrt{2^2} = 2
\]
Thus, \( SV = 2 \).
### Step 3: Relate the distance to the semi-major axis (A) and eccentricity (e)
The distance \( SV \) can also be expressed as:
\[
SV = A - AE
\]
Where \( A \) is the semi-major axis and \( E \) is the eccentricity. We can rewrite this as:
\[
A(1 - E) = 2
\]
Substituting \( E = \frac{2}{3} \):
\[
A\left(1 - \frac{2}{3}\right) = 2 \implies A\left(\frac{1}{3}\right) = 2 \implies A = 6
\]
### Step 4: Find the value of \( B^2 \)
Using the relationship between eccentricity, semi-major axis, and semi-minor axis:
\[
E = \sqrt{1 - \frac{B^2}{A^2}}
\]
Substituting the known values:
\[
\frac{2}{3} = \sqrt{1 - \frac{B^2}{6^2}} \implies \frac{4}{9} = 1 - \frac{B^2}{36}
\]
Rearranging gives:
\[
\frac{B^2}{36} = 1 - \frac{4}{9} = \frac{5}{9} \implies B^2 = \frac{5}{9} \times 36 = 20
\]
### Step 5: Determine the center of the ellipse
The center (H, K) of the ellipse can be found using the coordinates of the focus and vertex. The center is halfway between the focus and the vertex. The x-coordinate of the center is:
\[
\text{Center} (H, K) = (3 + 1)/2 = 2, 0
\]
So, the center is at (2, 0).
### Step 6: Write the equation of the ellipse
The standard form of the ellipse centered at (H, K) is:
\[
\frac{(x - H)^2}{A^2} + \frac{(y - K)^2}{B^2} = 1
\]
Substituting \( H = 2 \), \( K = 0 \), \( A = 6 \), and \( B^2 = 20 \):
\[
\frac{(x - 2)^2}{36} + \frac{y^2}{20} = 1
\]
### Step 7: Find the coordinates of the other focus
The distance from the center to each focus is given by \( AE \):
\[
AE = 6 \times \frac{2}{3} = 4
\]
Since one focus is at (3, 0), the other focus will be:
\[
(2 + 4, 0) = (6, 0)
\]
### Final Answer
The equation of the ellipse is:
\[
\frac{(x - 2)^2}{36} + \frac{y^2}{20} = 1
\]
The coordinates of the other focus are:
\[
(6, 0)
\]
