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 focus, directrix, and eccentricity, we can follow these steps:

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

### Step 2: Calculate the perpendicular distance from the focus to the directrix
The formula for the perpendicular distance from a point (x₀, y₀) to a line Ax + By + C = 0 is given by:

\[ \text{Distance} = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

For the directrix x + y + 2 = 0, we have:
- A = 1, B = 1, C = 2
- Point (x₀, y₀) = (3, 4)

Substituting these values into the formula:

\[ \text{Distance} = \frac{|1(3) + 1(4) + 2|}{\sqrt{1^2 + 1^2}} = \frac{|3 + 4 + 2|}{\sqrt{2}} = \frac{9}{\sqrt{2}} = \frac{9\sqrt{2}}{2} \]

### Step 3: Relate the distance to the eccentricity
According to the definition of eccentricity:

\[ \text{Distance} = e \cdot a \]

Where \( a \) is the semi-major axis of the ellipse. Thus, we can write:

\[ \frac{9\sqrt{2}}{2} = \frac{1}{2} \cdot a \]

Solving for \( a \):

\[ a = \frac{9\sqrt{2}}{2} \cdot 2 = 9\sqrt{2} \]

### Step 4: Calculate the semi-minor axis \( b \)
Using the relationship between \( a \), \( b \), and \( e \):

\[ b = a \sqrt{1 - e^2} \]

Substituting the values:

\[ b = 9\sqrt{2} \sqrt{1 - \left(\frac{1}{2}\right)^2} = 9\sqrt{2} \sqrt{1 - \frac{1}{4}} = 9\sqrt{2} \sqrt{\frac{3}{4}} = 9\sqrt{2} \cdot \frac{\sqrt{3}}{2} = \frac{9\sqrt{6}}{2} \]

### Step 5: Calculate the length of the latus rectum
The length of the latus rectum \( L \) of an ellipse is given by:

\[ L = \frac{2b^2}{a} \]

First, calculate \( b^2 \):

\[ b^2 = \left(\frac{9\sqrt{6}}{2}\right)^2 = \frac{81 \cdot 6}{4} = \frac{486}{4} = \frac{243}{2} \]

Now substitute \( b^2 \) and \( a \) into the formula for \( L \):

\[ L = \frac{2 \cdot \frac{243}{2}}{9\sqrt{2}} = \frac{243}{9\sqrt{2}} = \frac{27}{\sqrt{2}} = \frac{27\sqrt{2}}{2} \]

### Final Answer
The length of the latus rectum is \( \frac{27\sqrt{2}}{2} \).

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