Find the equation, axis, LR and vertex of the parabola whose focus is (0,0) and directrix is x + y + 1 = 0

To find the equation, axis, length of the latus rectum (LR), and vertex of the parabola given the focus \((0, 0)\) and the directrix \(x + y + 1 = 0\), we will proceed step by step.

### Step 1: Equation of the Parabola
The general formula for a parabola is derived based on the fact that the distance between any point \((x, y)\) on the parabola and the focus is equal to the perpendicular distance from that point to the directrix.

- **Focus**: \((0, 0)\)
- **Directrix**: \(x + y + 1 = 0\)

First, write the equation of the directrix in slope-intercept form:
\[
x + y + 1 = 0 \quad \Rightarrow \quad y = -x - 1
\]

The distance from any point \((x, y)\) on the parabola to the focus \((0, 0)\) is:
\[
\sqrt{(x - 0)^2 + (y - 0)^2} = \sqrt{x^2 + y^2}
\]

The perpendicular distance from the point \((x, y)\) to the directrix \(x + y + 1 = 0\) is given by:
\[
\frac{|x + y + 1|}{\sqrt{1^2 + 1^2}} = \frac{|x + y + 1|}{\sqrt{2}}
\]

Using the definition of the parabola (distance from the point to the focus equals distance to the directrix):
\[
\sqrt{x^2 + y^2} = \frac{|x + y + 1|}{\sqrt{2}}
\]

### Step 2: Simplify the Equation
Squaring both sides to eliminate the square root:
\[
x^2 + y^2 = \frac{(x + y + 1)^2}{2}
\]

Now expand the right-hand side:
\[
x^2 + y^2 = \frac{x^2 + y^2 + 2xy + 2x + 2y + 1}{2}
\]

Multiply both sides by 2 to simplify:
\[
2x^2 + 2y^2 = x^2 + y^2 + 2xy + 2x + 2y + 1
\]

Move all terms to one side:
\[
x^2 + y^2 - 2xy - 2x - 2y - 1 = 0
\]

This is the equation of the parabola.

### Step 3: Vertex of the Parabola
The vertex is the midpoint between the focus and the directrix. The directrix equation \(x + y + 1 = 0\) has a slope of -1. To find the point on the directrix closest to the focus, we use the formula for the perpendicular distance from a point to a line.

The closest point on the directrix to the focus \((0, 0)\) is:
\[
\frac{-1(0) + 0 + 1}{\sqrt{1^2 + (-1)^2}} = \frac{1}{\sqrt{2}} = \frac{1}{2}
\]
So the vertex will be at \((-\frac{1}{2}, -\frac{1}{2})\).

### Step 4: Axis of the Parabola
The axis of the parabola is perpendicular to the directrix. Since the slope of the directrix is \(-1\), the axis will have a slope of \(1\), making the axis along the line \(y = x\).

### Step 5: Length of the Latus Rectum (LR)
The length of the latus rectum is given by \(4p\), where \(p\) is the distance from the vertex to the focus (or the directrix). Here, \(p = \frac{\sqrt{2}}{2}\), so the length of the latus rectum is:
\[
LR = 4 \times \frac{\sqrt{2}}{2} = 2\sqrt{2}
\]

### Summary:
- **Equation** of the parabola: \(x^2 + y^2 - 2xy - 2x - 2y - 1 = 0\)
- **Vertex**: \(\left( -\frac{1}{2}, -\frac{1}{2} \right)\)
- **Axis**: Along the line \(y = x\)
- **Length of the Latus Rectum (LR)**: \(2\sqrt{2}\)

Would you like further details on any of the steps or clarifications?

Here are 5 related questions to expand on this:
1. How do you find the equation of a parabola given a focus and a directrix?
2. How does the position of the vertex relate to the focus and directrix?
3. How can the length of the latus rectum help describe the shape of a parabola?
4. What is the significance of the axis of a parabola?
5. How do you determine the orientation of a parabola in the coordinate plane?

**Tip**: Always remember, for a parabola, the distance between any point on the curve and the focus equals the perpendicular distance to the directrix.

Discover how to find the equation, axis, vertex, and length of the latus rectum (LR) of a parabola with focus (0, 0) and directrix x + y + 1 = 0. This problem covers key concepts in coordinate geometry and the properties of parabolas. Suitable for advanced high school or early college students.

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