The parametric coordinates of a point on the ellipse, whose foci are `(-3, 0)` and (9, 0) and eccentricity `(1)/(3)` , are

Options

A. `(-3 + 9 cos theta, 9 sin theta)`
B. `(4-3 cos theta, 4 + 9 sin theta)`
C. `( 3 + 18 cos theta, 4 + 9 sin theta)`
D. `( 3 + 18 cos theta, 12sqrt(2) sin theta)`

Answer

To find the parametric coordinates of a point on the ellipse with given foci and eccentricity, we can follow these steps:

### Step 1: Identify the foci and calculate the distance between them.
The foci of the ellipse are given as (-3, 0) and (9, 0). The distance between the foci can be calculated as:
\[ \text{Distance} = |9 - (-3)| = 9 + 3 = 12 \]

### Step 2: Relate the distance between foci to the semi-major axis \(a\).
The distance between the foci is equal to \(2c\), where \(c\) is the distance from the center to each focus. Thus, we have:
\[ 2c = 12 \implies c = 6 \]

### Step 3: Use the eccentricity to find the semi-major axis \(a\).
The eccentricity \(e\) is given as \(\frac{1}{3}\). The relationship between \(c\), \(a\), and \(e\) is:
\[ e = \frac{c}{a} \implies a = \frac{c}{e} = \frac{6}{\frac{1}{3}} = 18 \]

### Step 4: Calculate the semi-minor axis \(b\).
Using the relationship \(c^2 = a^2 - b^2\):
\[ c^2 = 6^2 = 36 \quad \text{and} \quad a^2 = 18^2 = 324 \]
Thus,
\[ 36 = 324 - b^2 \implies b^2 = 324 - 36 = 288 \implies b = \sqrt{288} = 12\sqrt{2} \]

### Step 5: Find the center of the ellipse.
The center of the ellipse is the midpoint of the foci:
\[ \text{Center} = \left( \frac{-3 + 9}{2}, 0 \right) = \left( 3, 0 \right) \]

### Step 6: Write the equation of the ellipse.
The standard form of the equation of the ellipse centered at (h, k) is:
\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \]
Substituting \(h = 3\), \(k = 0\), \(a^2 = 324\), and \(b^2 = 288\):
\[ \frac{(x - 3)^2}{324} + \frac{y^2}{288} = 1 \]

### Step 7: Determine the parametric coordinates.
The parametric equations for the ellipse are given by:
\[ x = h + a \cos \theta \quad \text{and} \quad y = k + b \sin \theta \]
Substituting the values:
\[ x = 3 + 18 \cos \theta \]
\[ y = 0 + 12\sqrt{2} \sin \theta \]

### Final Parametric Coordinates:
Thus, the parametric coordinates of a point on the ellipse are:
\[ (x, y) = (3 + 18 \cos \theta, 12\sqrt{2} \sin \theta) \]

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