Find the equation of the ellipse, whose centre is at (2, -3), one focus at (3, -3) and vertex at (4, -3).

Answer

To find the equation of the ellipse with the given parameters, we will follow these steps:

### Step 1: Identify the center, focus, and vertex
The center of the ellipse is given as (2, -3). The focus is at (3, -3) and the vertex is at (4, -3).

### Step 2: Determine the orientation of the ellipse
Since both the focus and vertex have the same y-coordinate as the center, the ellipse is horizontally oriented.

### Step 3: Calculate the value of 'a'
The distance from the center to the vertex (which is 'a') can be calculated as:
\[ a = \text{Distance from center to vertex} = 4 - 2 = 2 \]

### Step 4: Calculate the value of 'c'
The distance from the center to the focus (which is 'c') can be calculated as:
\[ c = \text{Distance from center to focus} = 3 - 2 = 1 \]

### Step 5: Use the relationship between a, b, and c
For ellipses, the relationship between 'a', 'b', and 'c' is given by:
\[ c^2 = a^2 - b^2 \]
We already have:
- \( a = 2 \) so \( a^2 = 4 \)
- \( c = 1 \) so \( c^2 = 1 \)

Now we can find 'b':
\[ 1 = 4 - b^2 \]
\[ b^2 = 4 - 1 = 3 \]

### Step 6: Write the equation of the ellipse
The standard form of the equation of a horizontally oriented ellipse is:
\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \]
Substituting the values:
- \( h = 2 \)
- \( k = -3 \)
- \( a^2 = 4 \)
- \( b^2 = 3 \)

The equation becomes:
\[ \frac{(x - 2)^2}{4} + \frac{(y + 3)^2}{3} = 1 \]

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

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