Find the equation of the ellipse whose eccentricity is `1/2`, a focus is (2,3) and a directrix is x=7. Find the length of the major and minor axes of the ellipse .

Answer

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

### Step 1: Identify the given parameters
- Eccentricity (e) = 1/2
- Focus (F) = (2, 3)
- Directrix (D) = x = 7

### Step 2: Use the definition of the ellipse
The definition of an ellipse states that for any point \( P(x, y) \) on the ellipse, the ratio of the distance from \( P \) to the focus \( F \) and the distance from \( P \) to the directrix \( D \) is equal to the eccentricity \( e \).

Mathematically, this can be expressed as:
\[ \frac{d(P, F)}{d(P, D)} = e \]

### Step 3: Calculate the distances
1. **Distance from point \( P(x, y) \) to focus \( F(2, 3) \)**:
\[ d(P, F) = \sqrt{(x - 2)^2 + (y - 3)^2} \]

2. **Distance from point \( P(x, y) \) to the directrix \( D(x = 7) \)**:
The distance from a point to a vertical line \( x = k \) is given by \( |x - k| \).
\[ d(P, D) = |x - 7| \]

### Step 4: Set up the equation using the eccentricity
Substituting the distances into the equation:
\[ \frac{\sqrt{(x - 2)^2 + (y - 3)^2}}{|x - 7|} = \frac{1}{2} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ \sqrt{(x - 2)^2 + (y - 3)^2} = \frac{1}{2} |x - 7| \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides results in:
\[ (x - 2)^2 + (y - 3)^2 = \frac{1}{4} (x - 7)^2 \]

### Step 7: Expand both sides
Expanding both sides:
- Left side:
\[ (x - 2)^2 + (y - 3)^2 = (x^2 - 4x + 4) + (y^2 - 6y + 9) = x^2 + y^2 - 4x - 6y + 13 \]

- Right side:
\[ \frac{1}{4} (x - 7)^2 = \frac{1}{4} (x^2 - 14x + 49) = \frac{1}{4} x^2 - \frac{7}{2} x + \frac{49}{4} \]

### Step 8: Set the equation to zero
Setting the equation to zero:
\[ x^2 + y^2 - 4x - 6y + 13 - \left(\frac{1}{4} x^2 - \frac{7}{2} x + \frac{49}{4}\right) = 0 \]

### Step 9: Combine like terms
Multiply through by 4 to eliminate the fraction:
\[ 4x^2 + 4y^2 - 16x - 24y + 52 - x^2 + 14x - 49 = 0 \]
Combine like terms:
\[ 3x^2 + 4y^2 - 2x - 24y + 3 = 0 \]

### Step 10: Rearranging into standard form
Rearranging gives:
\[ 3(x^2 - \frac{2}{3}x) + 4(y^2 - 6y) = -3 \]
Completing the square for \( x \) and \( y \):
\[ 3\left(x - \frac{1}{3}\right)^2 - \frac{1}{3} + 4(y - 3)^2 - 36 = -3 \]
\[ 3\left(x - \frac{1}{3}\right)^2 + 4(y - 3)^2 = 33 \]

### Step 11: Divide by 33
\[ \frac{(x - \frac{1}{3})^2}{11} + \frac{(y - 3)^2}{\frac{33}{4}} = 1 \]

### Step 12: Identify lengths of axes
From the standard form:
- Semi-major axis \( a = \sqrt{\frac{33}{4}} = \frac{\sqrt{33}}{2} \)
- Semi-minor axis \( b = \sqrt{11} \)

### Step 13: Calculate lengths of major and minor axes
- Length of major axis = \( 2a = \sqrt{33} \)
- Length of minor axis = \( 2b = 2\sqrt{11} \)

### Final Answer
- Equation of the ellipse:
\[ \frac{(x - \frac{1}{3})^2}{11} + \frac{(y - 3)^2}{\frac{33}{4}} = 1 \]
- Length of the major axis: \( \sqrt{33} \)
- Length of the minor axis: \( 2\sqrt{11} \)