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Find the points on the surface 1x² + 1y² + 3x² = 1 at which the tangent plane is parallel to the plane 3x + 5y – 2z = 2. and

Find the points on the surface 1x² + 1y² + 3x² = 1 at which the tangent plane is parallel to the plane 3x + 5y – 2z = 2. and

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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Question
**Problem Statement:** Find the points on the surface \(1x^2 + 1y^2 + 3z^2 = 1\) at which the tangent plane is parallel to the plane \(3x + 5y - 2z = 2\). **Solution:** Let the points be \((\Box, \Box, \Box)\) and \((\Box, \Box, \Box)\). **Explanation:** To solve this problem, the goal is to find points on the ellipsoid given by the equation \(x^2 + y^2 + 3z^2 = 1\) where the tangent plane at these points has the same orientation as the plane described by \(3x + 5y - 2z = 2\). This involves finding the gradient of the ellipsoid and setting it proportional to the normal vector of the given plane. The result will yield specific points \((x, y, z)\) satisfying these conditions. **Note:** The image included is a mathematical problem with spaces for the solution. There are no graphs or diagrams.
Transcribed Image Text:**Problem Statement:** Find the points on the surface \(1x^2 + 1y^2 + 3z^2 = 1\) at which the tangent plane is parallel to the plane \(3x + 5y - 2z = 2\). **Solution:** Let the points be \((\Box, \Box, \Box)\) and \((\Box, \Box, \Box)\). **Explanation:** To solve this problem, the goal is to find points on the ellipsoid given by the equation \(x^2 + y^2 + 3z^2 = 1\) where the tangent plane at these points has the same orientation as the plane described by \(3x + 5y - 2z = 2\). This involves finding the gradient of the ellipsoid and setting it proportional to the normal vector of the given plane. The result will yield specific points \((x, y, z)\) satisfying these conditions. **Note:** The image included is a mathematical problem with spaces for the solution. There are no graphs or diagrams.
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