Given the quadric surface z = 3r² - 2y2 answer each of the following questions. (a) What type of quadric surface is it? How do you know? Provide the evidence supporting how you classified the quadric surface. (b) Determine the intersection points of the surface with the line r(t) = (3t, 2t, 19t).
Given the quadric surface z = 3r² - 2y2 answer each of the following questions. (a) What type of quadric surface is it? How do you know? Provide the evidence supporting how you classified the quadric surface. (b) Determine the intersection points of the surface with the line r(t) = (3t, 2t, 19t).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I really need help trying to solve this problem can you please do this the calculus way to answer parts A and B because I don't understand it please it has to be the calculus way
![# Understanding Quadric Surfaces – A Detailed Educational Guide
Given the quadric surface \( z = 3x^2 - 2y^2 \), answer each of the following questions:
### (a) What type of quadric surface is it? How do you know? Provide the evidence supporting how you classified the quadric surface.
**Explanation:**
In the given equation \( z = 3x^2 - 2y^2 \), we need to classify the type of quadric surface.
- A **quadric surface** is a second-degree algebraic surface given by the general equation:
\[
Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0
\]
- For the given surface \( z = 3x^2 - 2y^2 \):
- Rearrange to standard form: \( z - 3x^2 + 2y^2 = 0 \).
- Here, the coefficients A and B are positive and negative respectively, indicating a hyperboloid of one sheet.
- The term \( 3x^2 \) represents a paraboloid opening upwards along the x-axis.
- The term \( -2y^2 \) represents a paraboloid opening downwards along the y-axis.
Therefore, the given surface is a **hyperbolic paraboloid**, a saddle-shaped curve. This conclusion is drawn from the mix of positive and negative coefficients of \( x^2 \) and \( y^2 \).
### (b) Determine the intersection points of the surface with the line \( \mathbf{r}(t) = \langle 3t, 2t, 19t \rangle \).
**Explanation:**
To determine the intersection points:
1. **Parametric Form of the Line**:
\[
\mathbf{r}(t) = \langle 3t, 2t, 19t \rangle
\]
Here, each component represents:
- \( x = 3t \)
- \( y = 2t \)
- \( z = 19t \)
2. **Substitute these into the Surface Equation**:
Given: \( z = 3x^2 - 2y^2 \)
Substitute](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff39d4e73-8442-4223-8525-8f555980c50a%2Fb7c5bbf6-3b0b-48d3-9f78-af9543e78b3a%2Fqkn06fk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:# Understanding Quadric Surfaces – A Detailed Educational Guide
Given the quadric surface \( z = 3x^2 - 2y^2 \), answer each of the following questions:
### (a) What type of quadric surface is it? How do you know? Provide the evidence supporting how you classified the quadric surface.
**Explanation:**
In the given equation \( z = 3x^2 - 2y^2 \), we need to classify the type of quadric surface.
- A **quadric surface** is a second-degree algebraic surface given by the general equation:
\[
Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0
\]
- For the given surface \( z = 3x^2 - 2y^2 \):
- Rearrange to standard form: \( z - 3x^2 + 2y^2 = 0 \).
- Here, the coefficients A and B are positive and negative respectively, indicating a hyperboloid of one sheet.
- The term \( 3x^2 \) represents a paraboloid opening upwards along the x-axis.
- The term \( -2y^2 \) represents a paraboloid opening downwards along the y-axis.
Therefore, the given surface is a **hyperbolic paraboloid**, a saddle-shaped curve. This conclusion is drawn from the mix of positive and negative coefficients of \( x^2 \) and \( y^2 \).
### (b) Determine the intersection points of the surface with the line \( \mathbf{r}(t) = \langle 3t, 2t, 19t \rangle \).
**Explanation:**
To determine the intersection points:
1. **Parametric Form of the Line**:
\[
\mathbf{r}(t) = \langle 3t, 2t, 19t \rangle
\]
Here, each component represents:
- \( x = 3t \)
- \( y = 2t \)
- \( z = 19t \)
2. **Substitute these into the Surface Equation**:
Given: \( z = 3x^2 - 2y^2 \)
Substitute
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