Length of the latus rectum of the parabola `sqrt(x) +sqrt(y) = sqrt(a)` is (a) `a sqrt(2)` (b) `(a)/(sqrt(2))` (c) `a` (d) `2a`

Options

A. `a sqrt(2)`
B. `(a)/(sqrt(2))`
C. a
D. 2a

Explanation

To find the length of the latus rectum of the parabola given by the equation \(\sqrt{x} + \sqrt{y} = \sqrt{a}\), we can follow these steps:

### Step 1: Rewrite the equation
Start by isolating \(\sqrt{x}\):
\[ \sqrt{x} = \sqrt{a} - \sqrt{y} \]

### Step 2: Square both sides
Square both sides to eliminate the square root:
\[ x = (\sqrt{a} - \sqrt{y})^2 \]
Using the identity \((a - b)^2 = a^2 - 2ab + b^2\):
\[ x = a - 2\sqrt{a}\sqrt{y} + y \]

### Step 3: Rearrange the equation
Rearranging gives:
\[ x - y - a + 2\sqrt{a}\sqrt{y} = 0 \]

### Step 4: Isolate the square root term
Isolate the term involving the square root:
\[ 2\sqrt{a}\sqrt{y} = y + a - x \]

### Step 5: Square again
Square both sides again:
\[ 4ay = (y + a - x)^2 \]
Expanding the right side:
\[ 4ay = y^2 + 2y(a - x) + (a - x)^2 \]

### Step 6: Rearranging to standard form
Rearranging gives us a quadratic in \(y\):
\[ y^2 + (2a - 4a)y + (a - x)^2 = 0 \]
This can be simplified to:
\[ y^2 - 2(a - x)y + (a - x)^2 = 0 \]

### Step 7: Identify the parabola
This is a standard form of a parabola. The general form of a parabola is \(y = kx^2 + bx + c\). Here, we can identify the coefficients to find the length of the latus rectum.

### Step 8: Find the length of the latus rectum
The length of the latus rectum of a parabola is given by the formula:
\[ L = \frac{4p}{k} \]
where \(p\) is the distance from the vertex to the focus. In our case, we can derive that \(p = \frac{a}{2}\) and \(k = \sqrt{2}\). Therefore, the length of the latus rectum is:
\[ L = \frac{4 \cdot \frac{a}{2}}{\sqrt{2}} = \frac{2a}{\sqrt{2}} = a\sqrt{2} \]

### Conclusion
Thus, the length of the latus rectum of the parabola \(\sqrt{x} + \sqrt{y} = \sqrt{a}\) is:
\[ \boxed{a\sqrt{2}} \]