Consider the paraboloid z = 2x² + 3y² and the plane z = x + y + 30, which intersects the paraboloid in a curve C at (4,1,35) as shown in the figure to the right. Find the equation of the line tangent to C at the point (4,1,35). Proceed by completing parts (a) through (d) below. x=4-5t OA. y=1-15t z = 35-10t a. Find a vector normal to the plane at (4,1,35). Choose the correct answer below. O A. (1,-1,-1) O B. (-1,-1,1) O C. (1,1,30) b. Find a vector normal to the plane tangent to the paraboloid at (4,1,35). Choose the correct answer below. O A. (-16, -6,1) O B. (16,6,30) OC. (-16, -6,30) OD. (16,6,1) c. The line tangent to C at (4,1,35) is orthogonal to both normal vectors found in parts (a) and (b). Use this fact to find a direction vector for the tangent line. Choose the correct answer below. x=4-15t OB. y=1+5t z= 35+t O D. (4,1,35) O A. (5.-15, -10) O B. (-15,5,1) O C. (5.0,-10) OD. (0, -15,- 10) d. Knowing a point on the tangent line and the direction of the tangent line, write an equation of the tangent line in parametric form. Choose the correct answer below. x = 4+ 5t O C. y=1-15t z=35-10t (4,1,35) (4,1) x = 4+ 5t OD. y=1+ 15t z = 35+ 10t ZA
Consider the paraboloid z = 2x² + 3y² and the plane z = x + y + 30, which intersects the paraboloid in a curve C at (4,1,35) as shown in the figure to the right. Find the equation of the line tangent to C at the point (4,1,35). Proceed by completing parts (a) through (d) below. x=4-5t OA. y=1-15t z = 35-10t a. Find a vector normal to the plane at (4,1,35). Choose the correct answer below. O A. (1,-1,-1) O B. (-1,-1,1) O C. (1,1,30) b. Find a vector normal to the plane tangent to the paraboloid at (4,1,35). Choose the correct answer below. O A. (-16, -6,1) O B. (16,6,30) OC. (-16, -6,30) OD. (16,6,1) c. The line tangent to C at (4,1,35) is orthogonal to both normal vectors found in parts (a) and (b). Use this fact to find a direction vector for the tangent line. Choose the correct answer below. x=4-15t OB. y=1+5t z= 35+t O D. (4,1,35) O A. (5.-15, -10) O B. (-15,5,1) O C. (5.0,-10) OD. (0, -15,- 10) d. Knowing a point on the tangent line and the direction of the tangent line, write an equation of the tangent line in parametric form. Choose the correct answer below. x = 4+ 5t O C. y=1-15t z=35-10t (4,1,35) (4,1) x = 4+ 5t OD. y=1+ 15t z = 35+ 10t ZA
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 #9
Transcribed Image Text:Consider the paraboloid z = 2x² + 3y² and the plane z = x + y + 30, which intersects the paraboloid in a curve C at (4,1,35) as shown in the figure to the
right. Find the equation of the line tangent to C at the point (4,1,35). Proceed by completing parts (a) through (d) below.
C
x = 4-5t
O A. y=1-15t
z = 35-10t
x=4-15t
OB. y=1+5t
z = 35+t
a. Find a vector normal to the plane at (4,1,35). Choose the correct answer below.
O A. (1,-1,-1)
O B. (-1,-1,1)
O C. (1,1,30)
b. Find a vector normal to the plane tangent to the paraboloid at (4,1,35). Choose the correct answer below.
O A. (-16, -6,1)
O D. (16,6,1)
O B. (16,6,30)
O C. (-16, -6,30)
c. The line tangent to C at (4,1,35) is orthogonal to both normal vectors found in parts (a) and (b). Use this fact to find a direction vector for the tangent line. Choose the correct answer
below.
O D. (4,1,35)
O A. (5.-15,-10)
O B. (-15,5,1)
O C. (5.0,- 10)
O D. (0, -15, 10)
d. Knowing a point on the tangent line and the direction of the tangent line, write an equation of the tangent line in parametric form. Choose the correct answer below.
x = 4+ 5t
O C. y=1-15t
z 35-10t
(4,1,35)
(4,1)
x = 4+ 5t
OD. y 1+ 15t
z = 35+ 10t
ZA
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