An ellipse has directrix `x+y=-2` focus at (3,4) eccentricity =1/2, then length of latus rectum is

Options

A. `(5)/(2)`
B. `(9)/(sqrt(2))`
C. `5 sqrt(2)`
D. none of these

Answer

To find the length of the latus rectum of the ellipse given the directrix, focus, and eccentricity, we can follow these steps:

### Step 1: Identify the given values
- Focus (F) = (3, 4)
- Directrix: x + y = -2
- Eccentricity (e) = 1/2

### Step 2: Find the distance from the focus to the directrix
The distance (d) from a point (x₁, y₁) to a line Ax + By + C = 0 is given by the formula:
\[ d = \frac{|Ax₁ + By₁ + C|}{\sqrt{A^2 + B^2}} \]
For the directrix x + y + 2 = 0, we have A = 1, B = 1, and C = 2. The coordinates of the focus are (3, 4).

Substituting these values into the formula:
\[ d = \frac{|1(3) + 1(4) + 2|}{\sqrt{1^2 + 1^2}} = \frac{|3 + 4 + 2|}{\sqrt{2}} = \frac{9}{\sqrt{2}} \]

### Step 3: Relate the distance to the eccentricity
For an ellipse, the relationship between the distance from the focus to the directrix (d), the semi-major axis (a), and the eccentricity (e) is given by:
\[ d = \frac{a}{e} - ae \]
Substituting the values we know:
\[ \frac{9}{\sqrt{2}} = \frac{a}{\frac{1}{2}} - a \cdot \frac{1}{2} \]
This simplifies to:
\[ \frac{9}{\sqrt{2}} = 2a - \frac{a}{2} \]
Combining the terms gives:
\[ \frac{9}{\sqrt{2}} = \frac{4a - a}{2} = \frac{3a}{2} \]

### Step 4: Solve for 'a'
To find 'a', we can rearrange the equation:
\[ 3a = \frac{9 \cdot 2}{\sqrt{2}} \implies a = \frac{9 \cdot 2}{3\sqrt{2}} = \frac{6}{\sqrt{2}} = 3\sqrt{2} \]

### Step 5: Find 'b²' using the eccentricity
We know that:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Substituting the value of e:
\[ \frac{1}{2} = \sqrt{1 - \frac{b^2}{(3\sqrt{2})^2}} = \sqrt{1 - \frac{b^2}{18}} \]
Squaring both sides:
\[ \frac{1}{4} = 1 - \frac{b^2}{18} \]
Rearranging gives:
\[ \frac{b^2}{18} = 1 - \frac{1}{4} = \frac{3}{4} \implies b^2 = 18 \cdot \frac{3}{4} = \frac{54}{4} = \frac{27}{2} \]

### Step 6: Calculate the length of the latus rectum
The length of the latus rectum (L) is given by:
\[ L = \frac{2b^2}{a} \]
Substituting the values of b² and a:
\[ L = \frac{2 \cdot \frac{27}{2}}{3\sqrt{2}} = \frac{27}{3\sqrt{2}} = \frac{9}{\sqrt{2}} \]

### Final Answer
The length of the latus rectum is:
\[ \frac{9}{\sqrt{2}} \]