https://file.aiquickdraw.com/mathbot/file/1729789380463tpo1vaso.jpg

Let's analyze and solve each statement in this problem step by step.

1. **Statement 1**:  
   "The plane \( 2x + 4y - z + 8 = 0 \) intersects the \(Ox\)-axis at the point \( \left( -\frac{1}{4}, 0, 0 \right) \)."

   To find the intersection with the \(x\)-axis, we set \(y = 0\) and \(z = 0\):
   \[
   2x + 8 = 0 \implies 2x = -8 \implies x = -4.
   \]
   So, the intersection point is \( (-4, 0, 0) \), not \( \left( -\frac{1}{4}, 0, 0 \right) \).  
   **This statement is incorrect.**

2. **Statement 2**:  
   "A plane always passes through any three arbitrary points, and only one such plane exists."

   This is true only if the three points are not collinear (i.e., they do not lie on the same line). In general, through any three non-collinear points, exactly one plane can be defined.  
   **This statement is correct, assuming non-collinear points.**

3. **Statement 3**:  
   "The plane \( x + 2y - z - 4 = 0 \) intersects the \(Oy\)-axis at the point \( (0, 2, 0) \)."

   To find the intersection with the \(y\)-axis, we set \(x = 0\) and \(z = 0\):
   \[
   2y - 4 = 0 \implies 2y = 4 \implies y = 2.
   \]
   Thus, the intersection point is indeed \( (0, 2, 0) \).  
   **This statement is correct.**

4. **Statement 4**:  
   "The plane \( 2x + 1 = 0 \) is parallel to the coordinate plane \(YOZ\)."

   The plane equation \(2x + 1 = 0\) can be rewritten as \(x = -\frac{1}{2}\). This is a vertical plane parallel to both the \(YOZ\)-plane, where \(x = 0\), and all planes with constant \(x\)-values.  
   **This statement is correct.**

### **Correct Answers:**
- Statement 2
- Statement 3
- Statement 4

Do you have any questions or need more details?

---

Here are 5 follow-up questions to deepen your understanding:
1. How do we determine if a plane intersects a coordinate axis at a certain point?
2. Why is only one unique plane defined through three non-collinear points?
3. What is the difference between planes parallel to coordinate planes and planes intersecting coordinate axes?
4. How can we rewrite plane equations to analyze their geometric properties better?
5. How does setting variables to zero help us find intersection points with axes?

**Tip:** When working with plane equations, always substitute variables systematically to confirm intersections with coordinate axes or parallels with coordinate planes.

Укажите все верные утверждения:
1. Плоскость 2x + 4y - z + 8 = 0 пересекает ось Ox в точке (−1/4; 0; 0).
2. Через произвольные три точки всегда проходит плоскость, и притом только одна.
3. Плоскость x + 2y - z - 4 = 0 пересекает ось Oy в точке (0; 2; 0).
4. Плоскость 2x + 1 = 0 параллельна координатной плоскости YOZ.

Analyze the truth of statements about planes intersecting coordinate axes and the properties of planes in 3D. This problem covers key concepts in analytic geometry and the intersection of planes with coordinate axes. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation