The latus ractum of a parabola whose directrix is `x+y -2=0` and focus is `(3,-4),` is

Options

A. `2 sqrt2`
B. `3sqrt2`
C. `6 sqrt2`
D. `(3)/(sqrt2)`

Answer

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

### Step 1: Identify the given information
- The directrix of the parabola is given by the equation:
\[ x + y - 2 = 0 \]
- The focus of the parabola is given as:
\[ (3, -4) \]

### Step 2: Calculate the distance between the focus and the directrix
The distance \(d\) from a point \((x_0, y_0)\) to a line \(Ax + By + C = 0\) can be calculated using the formula:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
For our directrix \(x + y - 2 = 0\), we have:
- \(A = 1\)
- \(B = 1\)
- \(C = -2\)

Substituting the coordinates of the focus \((3, -4)\):
\[ d = \frac{|1 \cdot 3 + 1 \cdot (-4) - 2|}{\sqrt{1^2 + 1^2}} = \frac{|3 - 4 - 2|}{\sqrt{2}} = \frac{|-3|}{\sqrt{2}} = \frac{3}{\sqrt{2}} \]

### Step 3: Relate the distance to the parameter \(a\)
For a parabola, the distance between the focus and the directrix is given by:
\[ 2a = d \]
Thus, we have:
\[ 2a = \frac{3}{\sqrt{2}} \]
From this, we can solve for \(a\):
\[ a = \frac{3}{2\sqrt{2}} \]

### Step 4: Calculate the length of the latus rectum
The length of the latus rectum \(L\) of a parabola is given by:
\[ L = 4a \]
Substituting the value of \(a\):
\[ L = 4 \cdot \frac{3}{2\sqrt{2}} = \frac{12}{2\sqrt{2}} = \frac{6}{\sqrt{2}} = 3\sqrt{2} \]

### Final Answer
The length of the latus rectum of the parabola is:
\[ \boxed{3\sqrt{2}} \]