Find symmetric equations for the tangent line to the curve of intersection of the paraboloid zx +y and the ellipsoid 2x +y? +z? 76 at the point (2, 2, 8).
Find symmetric equations for the tangent line to the curve of intersection of the paraboloid zx +y and the ellipsoid 2x +y? +z? 76 at the point (2, 2, 8).
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 16 on the typed sheet. I also hand wrote it.
Transcribed Image Text:The image contains handwritten mathematical text that reads:
"1(c) Find symmetric equation for the tangent line to the curve of intersection of the paraboloid \( z = x^2 + y^2 \) and the ellipsoid \( 2x^2 + 4y^2 + z^2 = 76 \) at \( P(2, 2, 8) \)."
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![Certainly! Here's a transcription and explanation suitable for an educational website:
---
## Calculus and Analytic Geometry: Tangent Planes and Lines
### Problems:
1. **Find the equation of a tangent plane to the surface** at the specified point.
\[
5x^2 + 3y^2 + 8z^2, \quad (3, 6, 5)
\]
2. **Find symmetric equations for the tangent line** to the curve of intersection of the paraboloid \(z = x^2 + y^2\) and the ellipsoid \(2x + y + z = 7\) at the point \((2, 2, 8)\).
### Additional Exercise:
Consider the function \(f(x, y) = \sin(xy)\) at the point \((4, 0)\).
- **Gradient Calculation Example:**
\[
\nabla f(x, y) = y \cos(xy) \mathbf{i} + x \cos(xy) \mathbf{j}
\]
- **Further Problem:**
Find \(x\) for when \(f(x)\) is given by the integral:
\[
\int \cos(8) \, dt = 0 \cdot \cos(x \pm 0) + 1 \cdot \cos(0)
\]
---
### Explanation:
- **Equation of the Tangent Plane:** This involves finding the partial derivatives with respect to each variable and evaluating them at the given point to create the equation of the plane.
- **Symmetric Equations for Tangent Line:** To find these, you compute the direction vector at the point of intersection and use it to write the symmetric form. This usually involves solving a system of equations derived from setting derivatives equal to the normal vector of both surfaces.
### Notes:
- The tangent plane and line are foundational concepts in multivariable calculus, aiding in understanding how surfaces and curves behave locally.
- **Gradient Vector:** Offers direction and rate of fastest increase and is essential in defining the tangent line and plane.
Use these problems to practice setting up and solving equations involving geometrical objects in three-dimensional space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabc5aa9d-951a-40e1-86a3-f092042c4046%2Fbfcafd1e-3573-4c00-9f11-2a0e7c58fc52%2F6vj1qi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Here's a transcription and explanation suitable for an educational website:
---
## Calculus and Analytic Geometry: Tangent Planes and Lines
### Problems:
1. **Find the equation of a tangent plane to the surface** at the specified point.
\[
5x^2 + 3y^2 + 8z^2, \quad (3, 6, 5)
\]
2. **Find symmetric equations for the tangent line** to the curve of intersection of the paraboloid \(z = x^2 + y^2\) and the ellipsoid \(2x + y + z = 7\) at the point \((2, 2, 8)\).
### Additional Exercise:
Consider the function \(f(x, y) = \sin(xy)\) at the point \((4, 0)\).
- **Gradient Calculation Example:**
\[
\nabla f(x, y) = y \cos(xy) \mathbf{i} + x \cos(xy) \mathbf{j}
\]
- **Further Problem:**
Find \(x\) for when \(f(x)\) is given by the integral:
\[
\int \cos(8) \, dt = 0 \cdot \cos(x \pm 0) + 1 \cdot \cos(0)
\]
---
### Explanation:
- **Equation of the Tangent Plane:** This involves finding the partial derivatives with respect to each variable and evaluating them at the given point to create the equation of the plane.
- **Symmetric Equations for Tangent Line:** To find these, you compute the direction vector at the point of intersection and use it to write the symmetric form. This usually involves solving a system of equations derived from setting derivatives equal to the normal vector of both surfaces.
### Notes:
- The tangent plane and line are foundational concepts in multivariable calculus, aiding in understanding how surfaces and curves behave locally.
- **Gradient Vector:** Offers direction and rate of fastest increase and is essential in defining the tangent line and plane.
Use these problems to practice setting up and solving equations involving geometrical objects in three-dimensional space.
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