Find the equation of the parabola whose focus is at the origin, and whose directrix is the line `y-x=4`.Find also the length of the latus rectum, the equation of the axis, and the coordinates of the vertex.

Answer

To find the equation of the parabola with the focus at the origin (0,0) and the directrix given by the line \(y - x = 4\), we can follow these steps:

### Step 1: Rewrite the Directrix Equation
The directrix can be rewritten in standard form:
\[ y - x - 4 = 0 \implies y = x + 4 \]

### Step 2: Define the Point on the Parabola
Let \(P(x, y)\) be any point on the parabola. According to the definition of a parabola, the distance from the point \(P\) to the focus \(F(0, 0)\) is equal to the perpendicular distance from \(P\) to the directrix.

### Step 3: Calculate the Distance from \(P\) to the Focus
Using the distance formula, the distance \(PF\) from point \(P(x, y)\) to the focus \(F(0, 0)\) is:
\[ PF = \sqrt{(x - 0)^2 + (y - 0)^2} = \sqrt{x^2 + y^2} \]

### Step 4: Calculate the Perpendicular Distance from \(P\) to the Directrix
The distance \(PM\) from point \(P(x, y)\) to the directrix \(y - x - 4 = 0\) can be calculated using the formula for the distance from a point to a line:
\[ PM = \frac{|y - x - 4|}{\sqrt{1^2 + (-1)^2}} = \frac{|y - x - 4|}{\sqrt{2}} \]

### Step 5: Set the Distances Equal
According to the definition of a parabola:
\[ PF = PM \]
Thus, we have:
\[ \sqrt{x^2 + y^2} = \frac{|y - x - 4|}{\sqrt{2}} \]

### Step 6: Square Both Sides
Squaring both sides gives:
\[ x^2 + y^2 = \frac{(y - x - 4)^2}{2} \]

### Step 7: Clear the Fraction
Multiply both sides by 2:
\[ 2(x^2 + y^2) = (y - x - 4)^2 \]

### Step 8: Expand the Right Side
Expanding the right side:
\[ 2x^2 + 2y^2 = y^2 - 2xy + x^2 + 16 - 8y + 8x \]
Rearranging gives:
\[ 2x^2 + 2y^2 = y^2 + x^2 - 2xy + 16 - 8y + 8x \]

### Step 9: Combine Like Terms
Bringing all terms to one side:
\[ 2x^2 + 2y^2 - y^2 - x^2 + 2xy - 8x + 8y - 16 = 0 \]
This simplifies to:
\[ x^2 + y^2 + 2xy - 8x + 8y - 16 = 0 \]

### Step 10: Final Equation of the Parabola
Thus, the equation of the parabola is:
\[ x^2 + y^2 + 2xy - 8x + 8y - 16 = 0 \]

### Step 11: Find the Length of the Latus Rectum
The length of the latus rectum \(L\) is given by \(4a\), where \(a\) is the distance from the focus to the directrix. The distance from the focus (0,0) to the line \(y - x - 4 = 0\) is:
\[ d = \frac{|0 - 0 - 4|}{\sqrt{1^2 + (-1)^2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2} \]
Thus, \(2a = 2\sqrt{2}\) implies \(a = \sqrt{2}\) and therefore the length of the latus rectum is:
\[ L = 4a = 4\sqrt{2} \]

### Step 12: Find the Equation of the Axis
The axis of the parabola is the line through the focus that is perpendicular to the directrix. The slope of the directrix \(y = x + 4\) is 1, so the slope of the axis is -1. The equation of the axis is:
\[ x + y = k \]
Since it passes through the focus (0,0), we have \(k = 0\), so the equation of the axis is:
\[ x + y = 0 \]

### Step 13: Find the Coordinates of the Vertex
The vertex of the parabola is the midpoint between the focus and the directrix. The directrix can be represented as a point, say \(M(4, 0)\). The midpoint \(V\) between \(F(0, 0)\) and the foot of the perpendicular from the focus to the directrix can be calculated as:
\[ V = \left(\frac{0 + 4}{2}, \frac{0 + 0}{2}\right) = (2, 0) \]

### Summary of Results
- **Equation of the Parabola**: \(x^2 + y^2 + 2xy - 8x + 8y - 16 = 0\)
- **Length of the Latus Rectum**: \(4\sqrt{2}\)
- **Equation of the Axis**: \(x + y = 0\)
- **Coordinates of the Vertex**: \((-1, 1)\)