The axis of a parabola is along the line y = x and the distance of its vertex from origin is `sqrt2` and that from its focus is `2sqrt2`. If vertex and focus both lie in the first quadrant, then the equation of the parabola is

Options

A. `(x+y)^(2)=(x-y-2)`
B. `(x-y)^(2)=(x+y-2)`
C. `(x-y)^(2)=4(x+y-2)`
D. `(x-y)^(2)=8(x+y-2)`

Answer

To find the equation of the parabola with the given conditions, we will follow these steps:

### Step 1: Understand the problem
The parabola has its axis along the line \(y = x\). The vertex is at a distance of \(\sqrt{2}\) from the origin, and the focus is at a distance of \(2\sqrt{2}\) from the origin. Both the vertex and focus are in the first quadrant.

### Step 2: Determine the coordinates of the vertex
Since the vertex is at a distance of \(\sqrt{2}\) from the origin and lies on the line \(y = x\), we can express the coordinates of the vertex as \((a, a)\), where \(a\) is the distance from the origin.

Using the distance formula:
\[ \sqrt{a^2 + a^2} = \sqrt{2} \]
This simplifies to:
\[ \sqrt{2a^2} = \sqrt{2} \implies 2a^2 = 2 \implies a^2 = 1 \implies a = 1 \]
Thus, the vertex is at \((1, 1)\).

### Step 3: Determine the coordinates of the focus
The focus is at a distance of \(2\sqrt{2}\) from the origin. Using the same reasoning as before, we can express the coordinates of the focus as \((b, b)\):
\[ \sqrt{b^2 + b^2} = 2\sqrt{2} \]
This simplifies to:
\[ \sqrt{2b^2} = 2\sqrt{2} \implies 2b^2 = 8 \implies b^2 = 4 \implies b = 2 \]
Thus, the focus is at \((2, 2)\).

### Step 4: Find the equation of the parabola
The vertex form of a parabola with its axis along the line \(y = x\) can be expressed as:
\[ (x - h)^2 = 4p(y - k) \]
where \((h, k)\) is the vertex and \(p\) is the distance from the vertex to the focus.

Here, the vertex is \((1, 1)\) and the focus is \((2, 2)\). The distance \(p\) is:
\[ p = \sqrt{(2 - 1)^2 + (2 - 1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 5: Substitute into the equation
Substituting \(h = 1\), \(k = 1\), and \(p = \sqrt{2}\):
\[ (x - 1)^2 = 4\sqrt{2}(y - 1) \]

### Step 6: Rearranging the equation
To express this in standard form:
\[ (x - 1)^2 = 4\sqrt{2}y - 4\sqrt{2} \]
\[ (x - 1)^2 - 4\sqrt{2}y + 4\sqrt{2} = 0 \]

### Step 7: Final form of the equation
This can be rearranged as:
\[ (x - 1)^2 - 4\sqrt{2}y + 4\sqrt{2} = 0 \]

### Conclusion
The equation of the parabola is:
\[ (x - 1)^2 = 4\sqrt{2}(y - 1) \]