The equation of the ellipse whose focus is (2, 4), centre is (3, 4) and eccentricity is 1/2 is

Options

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

Answer

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

### Step 1: Identify the given values
- Focus (F) = (2, 4)
- Center (C) = (3, 4)
- Eccentricity (e) = 1/2

### Step 2: Determine the distance between the center and the focus
The distance between the center and the focus (denoted as 'c') can be calculated using the formula:
\[ c = \sqrt{(x_F - x_C)^2 + (y_F - y_C)^2} \]
Substituting the values:
\[ c = \sqrt{(2 - 3)^2 + (4 - 4)^2} = \sqrt{(-1)^2 + 0} = \sqrt{1} = 1 \]

### Step 3: Relate 'c' to 'a' and 'e'
The relationship between the distance 'c', semi-major axis 'a', and eccentricity 'e' is given by:
\[ c = e \cdot a \]
Substituting the known values:
\[ 1 = \frac{1}{2} \cdot a \]
To find 'a', we solve for 'a':
\[ a = 2 \]

### Step 4: Calculate \( a^2 \)
Now, we need to find \( a^2 \):
\[ a^2 = 2^2 = 4 \]

### Step 5: Use the eccentricity to find \( b^2 \)
The relationship between eccentricity, semi-major axis, and semi-minor axis is given by:
\[ e^2 = 1 - \frac{b^2}{a^2} \]
Substituting the known values:
\[ \left(\frac{1}{2}\right)^2 = 1 - \frac{b^2}{4} \]
This simplifies to:
\[ \frac{1}{4} = 1 - \frac{b^2}{4} \]
Rearranging gives:
\[ \frac{b^2}{4} = 1 - \frac{1}{4} = \frac{3}{4} \]
Thus, multiplying both sides by 4:
\[ b^2 = 3 \]

### Step 6: Write the equation of the ellipse
The standard form of the ellipse centered at (α, β) is:
\[ \frac{(x - \alpha)^2}{a^2} + \frac{(y - \beta)^2}{b^2} = 1 \]
Substituting the values we have:
- \( \alpha = 3 \)
- \( \beta = 4 \)
- \( a^2 = 4 \)
- \( b^2 = 3 \)

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

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