The general equation of a quadratic surface is given by: a₁x² + a₂y² + az² + axy + açxz + ayz + ax + ay+agz + a₁0 = 0 Given nine points on this surface, it may be possible to determine its equation (a) Show that if the nine points (x,. y.) for i=1, 2, 3..... 9 lie on this surface, and if they determine uniquely the equation of this surface, then its equation can be written in determinant form as: x² x y² z² y z 2 2 Z4X4Y4 X4Z4 xy 23 Xsys X525 x²y² ZX6Y6 X626 X727 can do xy xz yz x y z 1 X₁1 XIZI VIZI X1 Y₁ ZI 1 X2Y2 X222 Y2Z2 X2 Y2 Z2 1 X3Y3 X323 Y3Z3 X3 Y3 Z3 1 Y4Z4 X4 Y4 YsZs X5 y's Zs Y6Z6 X6 Y6 Y7Z7 Y₁ Y8Z8 Xs Vs Y9Z9 X9 y² 27 X7Y7 z X8 Y8 X8Z8 X9Y9 X9Z9 y z N NNNN85 Z4 1 26 Z7 Zs 9 Z9 1 1 || 0 (b) Use the result in part (a) to determine the equation of the quadric surface that passes through the points (1, 2, 3), (2, 1, 7), (0, 4, 6), (3, -1, 4), (3, 0, 11), (-1, 5, 8), (9, -8, 3), (4) 5, 3), and (-2, 6, 10). (c) Classify the type of quadric surface in (b).
The general equation of a quadratic surface is given by: a₁x² + a₂y² + az² + axy + açxz + ayz + ax + ay+agz + a₁0 = 0 Given nine points on this surface, it may be possible to determine its equation (a) Show that if the nine points (x,. y.) for i=1, 2, 3..... 9 lie on this surface, and if they determine uniquely the equation of this surface, then its equation can be written in determinant form as: x² x y² z² y z 2 2 Z4X4Y4 X4Z4 xy 23 Xsys X525 x²y² ZX6Y6 X626 X727 can do xy xz yz x y z 1 X₁1 XIZI VIZI X1 Y₁ ZI 1 X2Y2 X222 Y2Z2 X2 Y2 Z2 1 X3Y3 X323 Y3Z3 X3 Y3 Z3 1 Y4Z4 X4 Y4 YsZs X5 y's Zs Y6Z6 X6 Y6 Y7Z7 Y₁ Y8Z8 Xs Vs Y9Z9 X9 y² 27 X7Y7 z X8 Y8 X8Z8 X9Y9 X9Z9 y z N NNNN85 Z4 1 26 Z7 Zs 9 Z9 1 1 || 0 (b) Use the result in part (a) to determine the equation of the quadric surface that passes through the points (1, 2, 3), (2, 1, 7), (0, 4, 6), (3, -1, 4), (3, 0, 11), (-1, 5, 8), (9, -8, 3), (4) 5, 3), and (-2, 6, 10). (c) Classify the type of quadric surface in (b).
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 general equation of a quadratic surface is given by:
a₁x² + a₂y² + a₂z² + ²xy + ²x² + y² + ax + ay + a² + a₁0 = 0
10
Given nine points on this surface, it may be possible to determine its equation
(a) Show that if the nine points (x,. y.) for i = 1, 2, 3,..., 9 lie on this surface, and if they
determine uniquely the equation of this surface, then its equation can be written in
determinant form as:
z² xy
z7 X1Y1
z3
z3
x y z
x²
x²
x²
x²
an do an do co
XZ
XIZI
ZX6Y6
X22 X2Z2 Y2Z2 X2
X3Y3 X3Z3
V3Z3 X3
X4Y4
X4Z4
Y4Z4 X4
Y4
x3y²z² Xsys
XsZs
YsZs Xs Ys
X626
Y6Z6 X6 Y6 26
27 X7X7 X727
Y7Z7 X7 Y₁
Y8Z8
X8
Ys
1929 X9 Y9
zx8Y8 X8Z8
yz
VIZI XI
x y z X99 X929
1
1
Y2 22 1
3
Z3 1
Z4 1
Z5 1
1
Z7 1
Z8 1
Z9 1
N
y
Y₁ ZI
= = 0
(b) Use the result in part (a) to determine the equation of the quadric surface that passes
through the points (1, 2, 3), (2, 1, 7), (0. 4, 6). (3.-1, 4), (3, 0, 11), (-1, 5, 8), (9,-8, 3), (4.
5, 3), and (-2, 6, 10).
(c) Classify the type of quadric surface in (b).
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