The axis of a parabola is along the line y = x and its vertex and focus are in the first quadrant at distances `sqrt2,2sqrt2` respectively, from the origin. 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)=(x-y-2)`
D. `(x-y)^(2)=8(x+y-2)`

Explanation

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

### Step 1: Determine the coordinates of the vertex and focus
The vertex is at a distance of \(\sqrt{2}\) from the origin. Since the vertex lies on the line \(y = x\), we can denote the coordinates of the vertex as \((h_1, h_1)\).

Using the distance formula:
\[ \sqrt{h_1^2 + h_1^2} = \sqrt{2} \]
This simplifies to:
\[ \sqrt{2h_1^2} = \sqrt{2} \implies 2h_1^2 = 2 \implies h_1^2 = 1 \implies h_1 = 1 \quad (\text{since } h_1 > 0) \]
Thus, the vertex is at \((1, 1)\).

Next, the focus is at a distance of \(2\sqrt{2}\) from the origin. Denote the coordinates of the focus as \((h_2, h_2)\). Using the distance formula again:
\[ \sqrt{h_2^2 + h_2^2} = 2\sqrt{2} \]
This simplifies to:
\[ \sqrt{2h_2^2} = 2\sqrt{2} \implies 2h_2^2 = 8 \implies h_2^2 = 4 \implies h_2 = 2 \quad (\text{since } h_2 > 0) \]
Thus, the focus is at \((2, 2)\).

### Step 2: Determine the equation of the directrix
The directrix of the parabola is a line perpendicular to the axis of the parabola, which is along \(y = x\). The midpoint between the vertex and the focus is the vertex itself, so the directrix will pass through the origin and have the equation:
\[ y + x = 0 \quad \text{or} \quad y = -x \]

### Step 3: Use the definition of a parabola
The definition of a parabola states that any point \((x, y)\) on the parabola is equidistant from the focus and the directrix.

1. **Distance from the focus \((2, 2)\)**:
\[ d_1 = \sqrt{(x - 2)^2 + (y - 2)^2} \]

2. **Distance from the directrix \(y + x = 0\)**:
The distance from a point \((x, y)\) to the line \(Ax + By + C = 0\) is given by:
\[ d_2 = \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}} \]
Here, \(A = 1\), \(B = 1\), and \(C = 0\):
\[ d_2 = \frac{|x + y|}{\sqrt{1^2 + 1^2}} = \frac{|x + y|}{\sqrt{2}} \]

### Step 4: Set the distances equal
Setting \(d_1 = d_2\):
\[ \sqrt{(x - 2)^2 + (y - 2)^2} = \frac{|x + y|}{\sqrt{2}} \]
Squaring both sides:
\[ (x - 2)^2 + (y - 2)^2 = \frac{(x + y)^2}{2} \]

### Step 5: Expand and simplify
Expanding both sides:
\[ (x^2 - 4x + 4) + (y^2 - 4y + 4) = \frac{x^2 + 2xy + y^2}{2} \]
Combining terms:
\[ x^2 + y^2 - 4x - 4y + 8 = \frac{x^2 + 2xy + y^2}{2} \]
Multiplying through by 2 to eliminate the fraction:
\[ 2x^2 + 2y^2 - 8x - 8y + 16 = x^2 + 2xy + y^2 \]
Rearranging gives:
\[ x^2 + y^2 - 2xy - 8x - 8y + 16 = 0 \]

### Step 6: Rearranging to standard form
Rearranging the equation gives:
\[ x^2 - 2xy + y^2 - 8x - 8y + 16 = 0 \]

This can be factored or rearranged to find the standard form of the parabola.

### Final Equation
The final equation of the parabola is:
\[ (x - y)^2 = 8(x + y - 2) \]