Determine the vertex, focus, directrix, ends of latus rectum, and axis of symmetry of y^2 = -36x

To solve the problem and find the vertex, focus, directrix, ends of the latus rectum, and axis of symmetry of the parabola given by the equation \( y^2 = -36x \), let's follow these steps:

### 1. **Equation Form**:
The given equation is \( y^2 = -36x \).

This is a parabola that opens leftward (since the coefficient of \(x\) is negative). The standard form of a parabola that opens left or right is:
\[
(y - k)^2 = 4p(x - h)
\]
where \((h, k)\) is the vertex and \( p \) is the distance from the vertex to the focus (or the directrix).

### 2. **Rewriting in Standard Form**:
We can rewrite \( y^2 = -36x \) as:
\[
(y - 0)^2 = -36(x - 0)
\]
Comparing this with the standard form \( (y - k)^2 = 4p(x - h) \), we identify:
- \( h = 0 \)
- \( k = 0 \)
- \( 4p = -36 \)

### 3. **Find the Vertex**:
The vertex \((h, k)\) is at \((0, 0)\).

### 4. **Determine \( p \)**:
Since \( 4p = -36 \), we solve for \( p \):
\[
p = \frac{-36}{4} = -9
\]

### 5. **Focus**:
The focus of the parabola is located at a distance \( p \) units from the vertex in the direction the parabola opens. Since \( p = -9 \) and the parabola opens to the left, the focus is:
\[
\text{Focus: } (h + p, k) = (0 - 9, 0) = (-9, 0)
\]

### 6. **Directrix**:
The directrix is a vertical line located \( p \) units away from the vertex in the opposite direction of the focus. So, the directrix is:
\[
\text{Directrix: } x = h - p = 0 - (-9) = 9
\]

### 7. **Ends of Latus Rectum**:
The latus rectum is a line segment perpendicular to the axis of symmetry and passes through the focus. Its length is \( |4p| = 36 \).

Since the latus rectum is centered at the focus and is horizontal, the ends of the latus rectum are:
\[
\text{Ends of latus rectum: } \left(-9, \frac{36}{2}\right) = (-9, 18) \quad \text{and} \quad \left(-9, -\frac{36}{2}\right) = (-9, -18)
\]

### 8. **Axis of Symmetry**:
The axis of symmetry is a vertical line passing through the vertex (and the focus). Hence:
\[
\text{Axis of Symmetry: } y = 0
\]

### **Summary**:
- **Vertex**: \((0, 0)\)
- **Focus**: \((-9, 0)\)
- **Directrix**: \(x = 9\)
- **Ends of Latus Rectum**: \((-9, 18)\) and \((-9, -18)\)
- **Axis of Symmetry**: \(y = 0\)

Would you like any more details on how these were determined, or have any specific questions?

---

Here are 5 questions related to this topic:
1. How would the vertex, focus, and directrix change if the equation were \( y^2 = 36x \)?
2. What is the geometric significance of the latus rectum in a parabola?
3. How would you find the length of the latus rectum for a parabola that opens upwards or downwards?
4. Can you convert the equation \( y^2 = -36x \) into the general quadratic form \( ax^2 + by^2 + cx + dy + e = 0 \)?
5. How would the vertex form change if the parabola were shifted 3 units right and 2 units up?

**Tip:** The distance from the vertex to the focus \( p \) determines the "width" of the parabola's opening. The larger the \( |p| \), the wider the parabola.

Learn how to determine the vertex, focus, directrix, and ends of latus rectum for the parabola y^2 = -36x with a step-by-step solution. Understand the geometric properties and equations involved in analyzing this type of parabola.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation