if the coordinates of the centre , a foucs and adjacent vertex are `(2,-3),(3,-3) and (4,-3)` respectively , then the equation of the ellipse Is

Options

A. `((x-2)^(2))/(4)+((y-3)^(2))/(3)=1`
B. `((x-3)^(2))/(4)+((y-2)^(2))/(3)=1`
C. `((x-2)^(2))/(8)+((y+3)^(2))/(6)=1`
D. `((x+2)^(2))/(4)+((y+3)^(2))/(3)=1`

Answer

To find the equation of the ellipse given the coordinates of the center, focus, and adjacent vertex, we can follow these steps:

### Step 1: Identify the Coordinates
We have:
- Center \( C(2, -3) \)
- Focus \( F(3, -3) \)
- Adjacent Vertex \( V(4, -3) \)

### Step 2: Determine the Orientation of the Ellipse
Since all three points have the same y-coordinate (-3), the ellipse is oriented horizontally. Therefore, the major axis is parallel to the x-axis.

### Step 3: Write the General Equation of the Ellipse
The general equation of an ellipse centered at \( (h, k) \) with a horizontal major axis is given by:
\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \]
Here, \( (h, k) = (2, -3) \).

### Step 4: Calculate the Values of \( a \) and \( b \)
1. **Distance from Center to Focus (c)**:
The distance from the center \( C \) to the focus \( F \) is:
\[ c = |3 - 2| = 1 \]

2. **Distance from Center to Vertex (a)**:
The distance from the center \( C \) to the vertex \( V \) is:
\[ a = |4 - 2| = 2 \]

3. **Relationship between \( a \), \( b \), and \( c \)**:
For ellipses, the relationship is given by:
\[ c^2 = a^2 - b^2 \]
We already have \( a = 2 \) and \( c = 1 \):
\[ c^2 = 1^2 = 1 \quad \text{and} \quad a^2 = 2^2 = 4 \]
Plugging these into the equation:
\[ 1 = 4 - b^2 \implies b^2 = 4 - 1 = 3 \]

### Step 5: Substitute Values into the Ellipse Equation
Now we substitute \( h = 2 \), \( k = -3 \), \( a^2 = 4 \), and \( b^2 = 3 \) into the general equation:
\[ \frac{(x - 2)^2}{4} + \frac{(y + 3)^2}{3} = 1 \]

### Final Equation of the Ellipse
Thus, the equation of the ellipse is:
\[ \frac{(x - 2)^2}{4} + \frac{(y + 3)^2}{3} = 1 \]

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