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5. Consider the saddle surface r = y- 2, find the point where the tangent plane is parallel to x+y+z = 1.

5. Consider the saddle surface r = y- 2, find the point where the tangent plane is parallel to x+y+z = 1.

Advanced Engineering Mathematics
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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**Problem 5: Tangent Plane to a Saddle Surface** Consider the saddle surface defined by the equation \( x = y^2 - z^2 \). The task is to find the point on this surface where the tangent plane is parallel to the plane described by the equation \( x + y + z = 1 \). **Solution Outline:** 1. **Identify the Gradient Vectors:** - The gradient vector of the surface \( x = y^2 - z^2 \) is computed to find the normal vector at any given point. - The normal vector to the given plane \( x + y + z = 1 \) is \( \langle 1, 1, 1 \rangle \). 2. **Set the Gradient Equal to the Normal Vector:** - Solve for the point \((y, z)\) where the gradient of the surface becomes parallel to the vector \( \langle 1, 1, 1 \rangle \). 3. **Calculate and Verify:** - Verify that the tangent plane at the found point is indeed parallel to the given plane by comparing their normal vectors.
Transcribed Image Text:**Problem 5: Tangent Plane to a Saddle Surface** Consider the saddle surface defined by the equation \( x = y^2 - z^2 \). The task is to find the point on this surface where the tangent plane is parallel to the plane described by the equation \( x + y + z = 1 \). **Solution Outline:** 1. **Identify the Gradient Vectors:** - The gradient vector of the surface \( x = y^2 - z^2 \) is computed to find the normal vector at any given point. - The normal vector to the given plane \( x + y + z = 1 \) is \( \langle 1, 1, 1 \rangle \). 2. **Set the Gradient Equal to the Normal Vector:** - Solve for the point \((y, z)\) where the gradient of the surface becomes parallel to the vector \( \langle 1, 1, 1 \rangle \). 3. **Calculate and Verify:** - Verify that the tangent plane at the found point is indeed parallel to the given plane by comparing their normal vectors.
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