The left panel of Figure 3 shows an elliptic hyperboloid of two sheets given by 4x² - y² - 2² = 1. The middle panel shows the plane 2xy + z = k intersecting the two sheets, resulting in two space curves. With the plane removed for clarity, the right panel just shows the space curves and the hyperboloid. The magenta and navy space curves will be referred to as C₁ and C₂ respectively. -10 -10 -10 10 -10 -10 -10 10 Figure 3 The parametrisation used to plot these space curves is 10 -10 -10 0 10 212-4t+5 x(t) 4(t-2) y(t) z(t) t 3-4t 2(1-2) a) Find the domain of t for C₁ and C2. (1) (2) (3) b) Using equations (1), (2) and (3) as a guide, determine the general space curve parametrisation in terms of k for the case presented above. This doesn't need to involve the 3 equations subbed directly into the hyperboloid/plane equations, just i.e sub the plane equation into the hyperboloid equation, leaving a relationship involving x,y and k; or y,z and k. Also, find the particular value of k for the case presented above. c) Parametrically define the hyperboloid.
The left panel of Figure 3 shows an elliptic hyperboloid of two sheets given by 4x² - y² - 2² = 1. The middle panel shows the plane 2xy + z = k intersecting the two sheets, resulting in two space curves. With the plane removed for clarity, the right panel just shows the space curves and the hyperboloid. The magenta and navy space curves will be referred to as C₁ and C₂ respectively. -10 -10 -10 10 -10 -10 -10 10 Figure 3 The parametrisation used to plot these space curves is 10 -10 -10 0 10 212-4t+5 x(t) 4(t-2) y(t) z(t) t 3-4t 2(1-2) a) Find the domain of t for C₁ and C2. (1) (2) (3) b) Using equations (1), (2) and (3) as a guide, determine the general space curve parametrisation in terms of k for the case presented above. This doesn't need to involve the 3 equations subbed directly into the hyperboloid/plane equations, just i.e sub the plane equation into the hyperboloid equation, leaving a relationship involving x,y and k; or y,z and k. Also, find the particular value of k for the case presented above. c) Parametrically define the hyperboloid.
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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Transcribed Image Text:The left panel of Figure 3 shows an elliptic hyperboloid of two sheets given by
4x² - y² - 2² = 1.
The middle panel shows the plane 2x
space curves. With the plane removed for clarity, the right panel just shows the space curves and the
hyperboloid.The magenta and navy space curves will be referred to as C₁ and C₂ respectively.
y + z = k intersecting the two sheets, resulting in two
-10
-10
-10
10
10
10
10
10
Zo
-10
-10
0
10
Figure 3
The parametrisation used to plot these space curves is
2t2-4t+5
-10
10
Zo
10
-10
10
0
10
x(t)
y(t)
4(t - 2)
=
t
z(t)
=
3- 4t
2(t-2)
a) Find the domain of t for C₁ and C2
(1)
དྱེསྱེºེ
(2)
(3)
b) Using equations (1), (2) and (3) as a guide, determine the general space curve
parametrisation in terms of k for the case presented above. This doesn't need to
involve the 3 equations subbed directly into the hyperboloid/plane equations,
just i.e sub the plane equation into the hyperboloid equation, leaving a
relationship involving x,y and k; or y,z and k. Also, find the particular value of k for
the case presented above.
c) Parametrically define the hyperboloid.
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