Check that the point (1,-1,2) lies on the given surface. Then, viewing the surface as a level surface for a function f(a, y, z), find a vector normal to the surface and an equation for the tangent plane to the surface at (1,-1, 2). vector normal = tangent plane: 4x² - y² + 3z² = 15

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 Statement:**

Check that the point \((1, -1, 2)\) lies on the given surface. Then, viewing the surface as a level surface for a function \(f(x, y, z)\), find a vector normal to the surface and an equation for the tangent plane to the surface at \((1, -1, 2)\).

**Given Surface Equation:**

\[ 4x^2 - y^2 + 3z^2 = 15 \]

**Tasks:**

1. Verify if the point \((1, -1, 2)\) satisfies the surface equation.
2. Determine a vector normal to the surface at this point.
3. Write the equation for the tangent plane at this point.

**Results Required:**

- **Vector Normal:** [Provide your answer here]
- **Tangent Plane:** \[ z = [Provide your equation here] \]
Transcribed Image Text:**Problem Statement:** Check that the point \((1, -1, 2)\) lies on the given surface. Then, viewing the surface as a level surface for a function \(f(x, y, z)\), find a vector normal to the surface and an equation for the tangent plane to the surface at \((1, -1, 2)\). **Given Surface Equation:** \[ 4x^2 - y^2 + 3z^2 = 15 \] **Tasks:** 1. Verify if the point \((1, -1, 2)\) satisfies the surface equation. 2. Determine a vector normal to the surface at this point. 3. Write the equation for the tangent plane at this point. **Results Required:** - **Vector Normal:** [Provide your answer here] - **Tangent Plane:** \[ z = [Provide your equation here] \]
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