The equation of parabola whose latus rectum is 2 units, axis of line is x+y-2=0 and tangent at the vertex is x-y+4=0 is given by

Options

A. `(x+y-2)^(2)=4sqrt(2)(x-y+4)^(2)`
B. `(x-y-4)^(2)=4sqrt(2)(x+y-2)`
C. `(x+y-2)^(2)=2sqrt(2)(x-y+4)`
D. none

Answer

To find the equation of the parabola given the conditions in the question, we can follow these steps:

### Step 1: Determine the value of 'a'
The length of the latus rectum of a parabola is given as 2 units. The formula relating the length of the latus rectum (L) to 'a' (the distance from the vertex to the focus) is:
\[ L = 4a \]

Given that \( L = 2 \):
\[ 4a = 2 \]
\[ a = \frac{2}{4} = \frac{1}{2} \]

### Step 2: Identify the axis of the parabola
The axis of the parabola is given by the line equation \( x + y - 2 = 0 \). We can rewrite this in slope-intercept form:
\[ y = -x + 2 \]
This indicates that the axis of the parabola has a slope of -1.

### Step 3: Find the slope of the tangent at the vertex
The tangent at the vertex is given by the equation \( x - y + 4 = 0 \). Rewriting this gives:
\[ y = x + 4 \]
This line has a slope of 1.

### Step 4: Determine the orientation of the parabola
Since the axis of the parabola has a slope of -1 and the tangent at the vertex has a slope of 1, we can conclude that the parabola opens downwards and to the left.

### Step 5: Write the standard form of the parabola
The standard form of a parabola with a vertical axis is:
\[ (y - k)^2 = 4a(x - h) \]
Where (h, k) is the vertex of the parabola.

However, since the axis is not vertical or horizontal, we need to rotate the coordinate system. The general equation of a parabola rotated by an angle θ can be expressed in terms of the new coordinates (x', y').

### Step 6: Substitute values into the equation
Using the values we have:
- \( a = \frac{1}{2} \)
- The vertex can be determined from the tangent line, which intersects the axis of the parabola.

The equation of the parabola can be derived from the conditions given, leading to:
\[ (x + y - 2)^2 = 2\sqrt{2}(x - y + 4) \]

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