Solution: Let (x, y) be old coordinates and Cr', y') be the new coordinates. The relation bet? them can be given by the foremula: x= x² coxo-ylsine..? } @ y = x²sind tylcoxo. where I + the angle of rotation of the coondinde ances Given, 3x² + 4xy = ⇒ 3 ( x'coxo-ylsino)² + 4 (x²coxo-ylsino) (xlcino + ylcoxo) = 16 ⇒ 3 [x²²cox ³0 + y² sin ²20-2xly/coxo.sino I +4 [x²²coxo, sino + xy/cox ³0 - xly's in ²0-ylisino.cox0] =16 By taking out coefficients of same terms common we 3 cox²0 +4 coxo sino ) + y¹² (3sin ²0 - y eino.coxo) toxy ( - 6 coxo-sino + y cox³e - 4 sin ²0 ) gef x ¹² ( 2 16 - -6 coxo sino + y cox ³0-usin ²0 =0. =) -3. 2 coxo sino + 4(cox ²0-sin ²0) =0 2/ -3 sin 20 + 4 cox2 = 0 2) 3 sin 20 = year 20 7 sin20 C०५२७ In the question we are asked to eliminate So we have to make the coefficient of x'y/ term ZeRo. ५ 3 पाउ =) tan 20 = 1/3 16-0 the oxy term, ху =
Solution: Let (x, y) be old coordinates and Cr', y') be the new coordinates. The relation bet? them can be given by the foremula: x= x² coxo-ylsine..? } @ y = x²sind tylcoxo. where I + the angle of rotation of the coondinde ances Given, 3x² + 4xy = ⇒ 3 ( x'coxo-ylsino)² + 4 (x²coxo-ylsino) (xlcino + ylcoxo) = 16 ⇒ 3 [x²²cox ³0 + y² sin ²20-2xly/coxo.sino I +4 [x²²coxo, sino + xy/cox ³0 - xly's in ²0-ylisino.cox0] =16 By taking out coefficients of same terms common we 3 cox²0 +4 coxo sino ) + y¹² (3sin ²0 - y eino.coxo) toxy ( - 6 coxo-sino + y cox³e - 4 sin ²0 ) gef x ¹² ( 2 16 - -6 coxo sino + y cox ³0-usin ²0 =0. =) -3. 2 coxo sino + 4(cox ²0-sin ²0) =0 2/ -3 sin 20 + 4 cox2 = 0 2) 3 sin 20 = year 20 7 sin20 C०५२७ In the question we are asked to eliminate So we have to make the coefficient of x'y/ term ZeRo. ५ 3 पाउ =) tan 20 = 1/3 16-0 the oxy term, ху =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
TRANSCRIBE THE FOLLOWING TEXT IN DIGITAL FORMAT
![Solution:
Let (x, y) be old coordinates and Cr', y') be the new
coordinates.
The relation bet? them can be given by the foremula:
} (2)
x = x² coxo-ylsin
y = x'sino + ylcoxo
202
cohere
O + the angle of rotation of the coordinade Gris
Given, 3x2+
3x² + 4xy=
⇒ 3 (x'coxo-ylsine)³ +4 (x'coxo-ylsine) (alcino + ylcox) = 16
=> 3 [x²²0 ²³0 + y² ²³sin ²0-2xly/coxo.sino ]
+4 [x²²coxo, sino + xy² cox ³0 - xly'sin?o-ylisino.cox0] =16
By taking out coefficients of same terms Common we get
3coy ²0 +4 coxo sino ) + y ² (30in²0. - y eino.cox@)
toxy ( - 6 coxo-sino + y cogie - 4 sin ²0 )
2
16
7 sin20
८०१२०
2) tan 20 = 1/3
In the question we are
asked to eliminate
the xy term,
So we have to make the coefficient of x'y' ferm ZeRo.
- 6 coxo sino + y cop ²0-ysin ²0
=) -3. 2 coxo sino + y(cox ²0-sin ²0) =0
2/ -3 sin 20 + 4 cox20 = 0
2)
3 sin 20 = year 20
1
3
(a) Rotation equis
=0.
1
16
7 0 26.57
R
4'²
Putting the value of 0 in ev., we gef
(3 cox (26-57) +4 co9 (26.57) sin(26-57) -
+ y¹² (3 sin? (26.57) - 4 sin (26.57)-cox (26.57)) = 16,
2 2²² ( 2-4 +1-6) + y²² (0.6-1.6)=16
=) [4x1²²_y₁²=16]
पयार
(b)
(Ⓒ) Value of tan (20) = 1/3
(b) Equation in the n'y' plane is
You!? - y1² = 1
2
=16
0 = n²coxo-ylsing
y = x² sino + ylcogo.
coefficient of x'y' = -600g0. sino + year²0 - usin?@](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4be50db6-19e4-41fe-8b0c-6d619944736f%2Febff3cd6-71a9-45e8-b6c8-704403311bed%2Fuumdpj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Solution:
Let (x, y) be old coordinates and Cr', y') be the new
coordinates.
The relation bet? them can be given by the foremula:
} (2)
x = x² coxo-ylsin
y = x'sino + ylcoxo
202
cohere
O + the angle of rotation of the coordinade Gris
Given, 3x2+
3x² + 4xy=
⇒ 3 (x'coxo-ylsine)³ +4 (x'coxo-ylsine) (alcino + ylcox) = 16
=> 3 [x²²0 ²³0 + y² ²³sin ²0-2xly/coxo.sino ]
+4 [x²²coxo, sino + xy² cox ³0 - xly'sin?o-ylisino.cox0] =16
By taking out coefficients of same terms Common we get
3coy ²0 +4 coxo sino ) + y ² (30in²0. - y eino.cox@)
toxy ( - 6 coxo-sino + y cogie - 4 sin ²0 )
2
16
7 sin20
८०१२०
2) tan 20 = 1/3
In the question we are
asked to eliminate
the xy term,
So we have to make the coefficient of x'y' ferm ZeRo.
- 6 coxo sino + y cop ²0-ysin ²0
=) -3. 2 coxo sino + y(cox ²0-sin ²0) =0
2/ -3 sin 20 + 4 cox20 = 0
2)
3 sin 20 = year 20
1
3
(a) Rotation equis
=0.
1
16
7 0 26.57
R
4'²
Putting the value of 0 in ev., we gef
(3 cox (26-57) +4 co9 (26.57) sin(26-57) -
+ y¹² (3 sin? (26.57) - 4 sin (26.57)-cox (26.57)) = 16,
2 2²² ( 2-4 +1-6) + y²² (0.6-1.6)=16
=) [4x1²²_y₁²=16]
पयार
(b)
(Ⓒ) Value of tan (20) = 1/3
(b) Equation in the n'y' plane is
You!? - y1² = 1
2
=16
0 = n²coxo-ylsing
y = x² sino + ylcogo.
coefficient of x'y' = -600g0. sino + year²0 - usin?@
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