Equation of the parabola whose axis is `y=x` distance from origin to vertex is `sqrt2` and distance from origin to focus is `2sqrt2`, is (Focus and vertex lie in Ist quadrant):

Options

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

Answer

To find the equation of the parabola whose axis is \( y = x \), with the given distances from the origin to the vertex and focus, we can follow these steps:

### Step 1: Understand the properties of the parabola
The parabola opens along the line \( y = x \), which means its vertex and focus will also lie along this line. The distance from the origin (0, 0) to the vertex (V) is given as \( \sqrt{2} \) and the distance from the origin to the focus (F) is \( 2\sqrt{2} \).

### Step 2: Determine the coordinates of the vertex
Since the distance from the origin to the vertex is \( \sqrt{2} \), we can express the coordinates of the vertex \( V \) as:
\[ V = (a, a) \]
where \( a = \sqrt{2}/\sqrt{2} = 1 \) (since the vertex lies in the first quadrant). Thus, the coordinates of the vertex are:
\[ V = (1, 1) \]

### Step 3: Determine the coordinates of the focus
The distance from the origin to the focus is \( 2\sqrt{2} \). Since the focus also lies on the line \( y = x \), we can express its coordinates as:
\[ F = (b, b) \]
The distance from the origin to the focus is given by:
\[ \sqrt{(b - 0)^2 + (b - 0)^2} = 2\sqrt{2} \]
This simplifies to:
\[ \sqrt{2b^2} = 2\sqrt{2} \]
Squaring both sides:
\[ 2b^2 = 8 \implies b^2 = 4 \implies b = 2 \]
Thus, the coordinates of the focus are:
\[ F = (2, 2) \]

### Step 4: Determine the equation of the directrix
The distance between the vertex and the focus is equal to the distance from the vertex to the directrix. The distance between \( V \) and \( F \) is:
\[ \sqrt{(2-1)^2 + (2-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
Since the directrix is perpendicular to the axis of the parabola and lies at a distance of \( \sqrt{2} \) from the vertex, we can find its equation. The axis of the parabola is \( y = x \), so the directrix will be \( y = -x \).

### Step 5: Write the equation of the parabola
The general equation of a parabola with vertex at \( (h, k) \) and focus at \( (h + p, k + p) \) is:
\[ (x - h)^2 = 4p(y - k) \]
Here, \( h = 1 \), \( k = 1 \), and \( p = 1 \) (the distance from the vertex to the focus). Thus, the equation becomes:
\[ (x - 1)^2 = 4(y - 1) \]

### Step 6: Rearranging the equation
Rearranging gives us:
\[ (x - 1)^2 = 4y - 4 \]
or
\[ (x - 1)^2 - 4y + 4 = 0 \]

### Final Equation
Thus, the equation of the parabola is:
\[ (x - 1)^2 = 4(y - 1) \]