Find the parametric equations of the tangent line to the curve of intersection of the surfaces x² - y² + z² = 17, 5x - 4y -z = 0 at (2,3,-2).
Find the parametric equations of the tangent line to the curve of intersection of the surfaces x² - y² + z² = 17, 5x - 4y -z = 0 at (2,3,-2).
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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![### Finding the Parametric Equations of the Tangent Line
**Problem Statement:**
11. Find the parametric equations of the tangent line to the curve of intersection of the surfaces
\[ x^2 - y^2 + z^2 = 17 \]
\[ 5x - 4y - z = 0 \]
at the point \((2, 3, -2)\).
---
**Solution Outline:**
To find the parametric equations of the tangent line at the given point, we need to follow these steps:
1. Calculate the gradient vectors of the given surfaces at the point of intersection.
2. Find the direction vector of the tangent line by taking the cross product of the gradient vectors.
3. Write the parametric equations using the point and the direction vector.
**Step-by-Step Solution:**
1. **Calculate the gradient vectors:**
For the surface \( x^2 - y^2 + z^2 = 17 \):
\[ \nabla f = \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \]
\[
f(x,y,z) = x^2 - y^2 + z^2
\]
Therefore,
\[
\nabla f = (2x, -2y, 2z)
\]
Evaluating at \((2, 3, -2)\),
\[
\nabla f(2, 3, -2) = (4, -6, -4)
\]
For the surface \( 5x - 4y - z = 0 \):
\[ \nabla g = (5, -4, -1) \]
2. **Find the direction vector of the tangent line:**
We need to take the cross product of \(\nabla f\) and \(\nabla g\).
\[
\vec{d} = \nabla f \times \nabla g
\]
\[
\vec{d} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
4 & -6 & -4 \\
5 & -4 & -1
\end{vmatrix](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F42ddf74f-f146-428c-864f-8ecdeefbd630%2Fa8848edf-0b9b-4022-8289-a25d4ffde698%2Fql46wqd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Parametric Equations of the Tangent Line
**Problem Statement:**
11. Find the parametric equations of the tangent line to the curve of intersection of the surfaces
\[ x^2 - y^2 + z^2 = 17 \]
\[ 5x - 4y - z = 0 \]
at the point \((2, 3, -2)\).
---
**Solution Outline:**
To find the parametric equations of the tangent line at the given point, we need to follow these steps:
1. Calculate the gradient vectors of the given surfaces at the point of intersection.
2. Find the direction vector of the tangent line by taking the cross product of the gradient vectors.
3. Write the parametric equations using the point and the direction vector.
**Step-by-Step Solution:**
1. **Calculate the gradient vectors:**
For the surface \( x^2 - y^2 + z^2 = 17 \):
\[ \nabla f = \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \]
\[
f(x,y,z) = x^2 - y^2 + z^2
\]
Therefore,
\[
\nabla f = (2x, -2y, 2z)
\]
Evaluating at \((2, 3, -2)\),
\[
\nabla f(2, 3, -2) = (4, -6, -4)
\]
For the surface \( 5x - 4y - z = 0 \):
\[ \nabla g = (5, -4, -1) \]
2. **Find the direction vector of the tangent line:**
We need to take the cross product of \(\nabla f\) and \(\nabla g\).
\[
\vec{d} = \nabla f \times \nabla g
\]
\[
\vec{d} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
4 & -6 & -4 \\
5 & -4 & -1
\end{vmatrix
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