Let C be the curve of intersection of the spheres x² + y2 + z² = 83 and (x - 2)² + (y-2)² + z² = 83. Find the parametric equations of the tangent line to C at P = (1, 1,9). It is known that if the intersection of two surfaces F(x, y, z) = 0 and G(x, y, z) = 0 is a curve C and P is a point on C, then the vector v = VFp x VGp is a direction vector for the tangent line to C at P. (Use symbolic notation and fractions where needed. Enter your answers as functions of parameter t in a form r(t) = (x(t), y(t), z(t)) = ro + vt, where ro is the corresponding coordinate of point P.)
Let C be the curve of intersection of the spheres x² + y2 + z² = 83 and (x - 2)² + (y-2)² + z² = 83. Find the parametric equations of the tangent line to C at P = (1, 1,9). It is known that if the intersection of two surfaces F(x, y, z) = 0 and G(x, y, z) = 0 is a curve C and P is a point on C, then the vector v = VFp x VGp is a direction vector for the tangent line to C at P. (Use symbolic notation and fractions where needed. Enter your answers as functions of parameter t in a form r(t) = (x(t), y(t), z(t)) = ro + vt, where ro is the corresponding coordinate of point P.)
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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![**Parametric Equations of the Tangent Line at the Intersection of Spheres**
Let \( C \) be the curve of intersection of the spheres \( x^2 + y^2 + z^2 = 83 \) and \( (x-2)^2 + (y-2)^2 + z^2 = 83 \).
We are asked to find the parametric equations of the tangent line to \( C \) at \( P = (1, 1, 9) \).
It is known that if the intersection of two surfaces \( F(x, y, z) = 0 \) and \( G(x, y, z) = 0 \) is a curve \( C \) and \( P \) is a point on \( C \), then the vector \( \mathbf{v} = \nabla F_P \times \nabla G_P \) is a direction vector for the tangent line to \( C \) at \( P \).
**Use symbolic notation and fractions where needed. Enter your answers as functions of parameter \( t \) in a form \(\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle = \mathbf{r_0} + \mathbf{v}t\), where \(\mathbf{r_0}\) is the corresponding coordinate of point \( P \).**
### Solution:
Given:
- \( x^2 + y^2 + z^2 = 83 \)
- \( (x-2)^2 + (y-2)^2 + z^2 = 83 \)
- Point \( P = (1, 1, 9) \)
To find the parametric equations:
1. **Calculate Gradients**
- Gradient of \( F(x,y,z) \) at \( P \):
\[
F(x, y, z) = x^2 + y^2 + z^2 - 83
\]
\[
\nabla F = \left\langle 2x, 2y, 2z \right\rangle
\]
At point \( P = (1, 1, 9) \):
\[
\nabla F_P = \left\langle 2(1), 2(1), 2(9) \right\rangle = \left](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faccc7427-2c55-44a5-b68e-19581b7568be%2Ff7ae6b74-aa21-42cc-960b-4bc103064086%2Fl8198ja_processed.png&w=3840&q=75)
Transcribed Image Text:**Parametric Equations of the Tangent Line at the Intersection of Spheres**
Let \( C \) be the curve of intersection of the spheres \( x^2 + y^2 + z^2 = 83 \) and \( (x-2)^2 + (y-2)^2 + z^2 = 83 \).
We are asked to find the parametric equations of the tangent line to \( C \) at \( P = (1, 1, 9) \).
It is known that if the intersection of two surfaces \( F(x, y, z) = 0 \) and \( G(x, y, z) = 0 \) is a curve \( C \) and \( P \) is a point on \( C \), then the vector \( \mathbf{v} = \nabla F_P \times \nabla G_P \) is a direction vector for the tangent line to \( C \) at \( P \).
**Use symbolic notation and fractions where needed. Enter your answers as functions of parameter \( t \) in a form \(\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle = \mathbf{r_0} + \mathbf{v}t\), where \(\mathbf{r_0}\) is the corresponding coordinate of point \( P \).**
### Solution:
Given:
- \( x^2 + y^2 + z^2 = 83 \)
- \( (x-2)^2 + (y-2)^2 + z^2 = 83 \)
- Point \( P = (1, 1, 9) \)
To find the parametric equations:
1. **Calculate Gradients**
- Gradient of \( F(x,y,z) \) at \( P \):
\[
F(x, y, z) = x^2 + y^2 + z^2 - 83
\]
\[
\nabla F = \left\langle 2x, 2y, 2z \right\rangle
\]
At point \( P = (1, 1, 9) \):
\[
\nabla F_P = \left\langle 2(1), 2(1), 2(9) \right\rangle = \left
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