The length of the latusrectum of the parabola `2{(x-a)^(2)+(y-a)^(2)}=(x+y)^(2),` is

Options

A. 2a
B. `2sqrt(2)a`
C. 4a
D. `sqrt2a`

Explanation

To find the length of the latus rectum of the given parabola \( 2(x-a)^2 + (y-a)^2 = (x+y)^2 \), we will follow these steps:

### Step 1: Rewrite the given equation
We start with the equation:
\[ 2(x-a)^2 + (y-a)^2 = (x+y)^2 \]
This can be rearranged to express it in a more standard form.

### Step 2: Expand both sides
Expanding both sides gives:
\[ 2(x^2 - 2ax + a^2) + (y^2 - 2ay + a^2) = x^2 + 2xy + y^2 \]
This simplifies to:
\[ 2x^2 - 4ax + 2a^2 + y^2 - 2ay + a^2 = x^2 + 2xy + y^2 \]

### Step 3: Combine like terms
Combining like terms results in:
\[ 2x^2 - x^2 - 4ax - 2xy - 2ay + 3a^2 = 0 \]
This simplifies to:
\[ x^2 - 2xy - 4ax - 2ay + 3a^2 = 0 \]

### Step 4: Identify the focus and directrix
From the standard form of a parabola, we can identify the focus and directrix. The focus \( S \) is at \( (a, a) \) and the directrix \( D \) can be represented as the line \( x + y = 0 \).

### Step 5: Calculate the distance from the focus to the directrix
The distance \( d \) from the focus \( S(a, a) \) to the directrix \( x + y = 0 \) can be calculated using the formula for the distance from a point to a line:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
where \( A = 1, B = 1, C = 0 \) and \( (x_1, y_1) = (a, a) \):
\[ d = \frac{|1 \cdot a + 1 \cdot a + 0|}{\sqrt{1^2 + 1^2}} = \frac{2a}{\sqrt{2}} = a\sqrt{2} \]

### Step 6: Find the length of the latus rectum
The length of the latus rectum \( LR \) of a parabola is given by:
\[ LR = 2 \times d \]
Substituting the distance we found:
\[ LR = 2 \times a\sqrt{2} = 2a\sqrt{2} \]

### Final Answer
Thus, the length of the latus rectum of the parabola is:
\[ \boxed{2a\sqrt{2}} \]