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 of the parabola, its axis, the length of the latus rectum (LR), and its vertex, we use the given information:

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

### Step 1: Equation of the Parabola
The general form of a parabola is the set of points equidistant from the focus and the directrix. Let \( P(x, y) \) be any point on the parabola. The distance between \( P(x, y) \) and the focus \( (0, 0) \) is given by:

\[
\text{Distance to focus} = \sqrt{x^2 + y^2}
\]

The distance from \( P(x, y) \) to the directrix \( x + y + 1 = 0 \) is given by the formula for the distance from a point to a line:

\[
\text{Distance to directrix} = \frac{|x + y + 1|}{\sqrt{1^2 + 1^2}} = \frac{|x + y + 1|}{\sqrt{2}}
\]

Equating these two distances (since a parabola is defined as the locus of points equidistant from the focus and directrix):

\[
\sqrt{x^2 + y^2} = \frac{|x + y + 1|}{\sqrt{2}}
\]

### Step 2: Squaring both sides
Square both sides of the equation to eliminate the square root:

\[
x^2 + y^2 = \frac{(x + y + 1)^2}{2}
\]

Now, expand the right-hand side:

\[
x^2 + y^2 = \frac{(x^2 + 2xy + y^2 + 2x + 2y + 1)}{2}
\]

Multiply both sides by 2 to eliminate the denominator:

\[
2(x^2 + y^2) = x^2 + 2xy + y^2 + 2x + 2y + 1
\]

### Step 3: Simplifying
Move all terms to one side:

\[
2x^2 + 2y^2 - x^2 - y^2 - 2xy - 2x - 2y - 1 = 0
\]

Simplify the terms:

\[
x^2 + y^2 - 2xy - 2x - 2y - 1 = 0
\]

This is the equation of the parabola.

### Step 4: Finding the Axis, Vertex, and Length of the Latus Rectum (LR)
- **Axis**: The axis of the parabola will be perpendicular to the directrix. The slope of the directrix \( x + y + 1 = 0 \) is \( -1 \), so the axis of symmetry has a slope of 1. Therefore, the equation of the axis is along the line \( y = x \).

- **Vertex**: The vertex lies midway between the focus \( (0, 0) \) and the directrix. The distance from the focus to the directrix is:

\[
\frac{|0 + 0 + 1|}{\sqrt{2}} = \frac{1}{\sqrt{2}}
\]

The midpoint between the focus and the directrix (along the axis of symmetry) gives the vertex. This lies on the line \( y = x \). The vertex is therefore at:

\[
\left( -\frac{1}{2\sqrt{2}}, -\frac{1}{2\sqrt{2}} \right)
\]

- **Latus Rectum (LR)**: The length of the latus rectum of a parabola is \( |4p| \), where \( p \) is the distance from the vertex to the focus. In this case, the distance \( p \) is \( \frac{1}{\sqrt{2}} \), so the latus rectum is:

\[
\text{LR} = 4 \times \frac{1}{\sqrt{2}} = 2\sqrt{2}
\]

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

Would you like any further clarification on the steps? 

Here are 5 related questions for further exploration:
1. How would the equation change if the focus were moved to \( (1, 1) \)?
2. What if the directrix was \( x - y + 1 = 0 \)? How would that affect the equation?
3. How do you find the equation of a parabola given its vertex and focus?
4. What is the geometric significance of the latus rectum in a parabola?
5. How does the orientation of the directrix affect the shape and orientation of the parabola?

**Tip**: For a parabola, the focus and directrix always determine the vertex, axis, and other geometric properties.

Learn how to find the equation, axis, latus rectum (LR), and vertex of a parabola with a focus at (0,0) and a directrix given by x + y + 1 = 0. This problem involves using the definition of a parabola, applying distance formulas, and simplifying the equation for Grades 10-12 students.

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