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

Explanation

To find the equation of the parabola whose axis is along the line \( y = x \), and given that the distances of its vertex and focus from the origin are \( \sqrt{2} \) and \( 2\sqrt{2} \) respectively, we can follow these steps:

### Step 1: Identify the coordinates of the vertex and focus
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 \( a = \sqrt{2}/\sqrt{2} = 1 \). Thus, the vertex is at \( (1, 1) \).

### Step 2: Identify the coordinates of the focus
The focus is at a distance of \( 2\sqrt{2} \) from the origin, and since it also lies on the line \( y = x \), we can express its coordinates as \( (b, b) \). The distance from the origin is given by:
\[ \sqrt{b^2 + b^2} = \sqrt{2b^2} = 2\sqrt{2} \]
Squaring both sides gives:
\[ 2b^2 = 8 \implies b^2 = 4 \implies b = 2 \]
Thus, the focus is at \( (2, 2) \).

### Step 3: Determine the equation of the directrix
The directrix of a parabola is located at a distance equal to the distance from the vertex to the focus, but in the opposite direction. The distance from the vertex \( (1, 1) \) to the focus \( (2, 2) \) is:
\[ \sqrt{(2-1)^2 + (2-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
Thus, the directrix will be a line perpendicular to the axis of the parabola (which is along \( y = x \)) and passing through the vertex. The equation of the directrix can be derived from the line \( y = -x + c \) where \( c \) is a constant.

To find \( c \), we can use the distance from the vertex to the directrix, which should also be \( \sqrt{2} \):
\[ \frac{|1 + 1 - c|}{\sqrt{2}} = \sqrt{2} \]
This simplifies to:
\[ |2 - c| = 2 \implies c = 0 \text{ or } c = 4 \]
Since we want the directrix to be below the vertex, we take \( c = 0 \), giving us the directrix:
\[ x + y = 0 \]

### Step 4: Use the definition of a parabola
The definition of a parabola states that the distance from any point \( P(x, y) \) on the parabola to the focus \( (2, 2) \) is equal to the distance from \( P(x, y) \) to the directrix \( x + y = 0 \).

Thus, we can write:
\[ \sqrt{(x - 2)^2 + (y - 2)^2} = \frac{|x + y|}{\sqrt{2}} \]

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

### Final Equation
Thus, the equation of the parabola is:
\[ x^2 + y^2 - 2xy - 8x - 8y + 16 = 0 \]