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Translation and Rotation of Axes Worksheet - Page 3

Translation and Rotation of Axes Worksheet
  • Page 3
 21.  
Find the discriminant of the tranformed equation of Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 when the axes are translated to (h, k).
a.
B2 - 2AC
b.
B2 + 4AC
c.
B2 - AC
d.
B2 - 4AC


Solution:

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
[Original equation.]

For the above equation, Discriminant = B2 - 4AC.

x = x′ + h, y = y′ + k
[Axes are translated to (h, k).]

A(x′ + h)2 + B(x′ + h) (y′+ k) + C(y′ + k)2 + D(x′ + h) + E(y′ + k) + F = 0
[Substitute x = x′ + h, y = y′ + k.]

A(x′)2 + Bxy′ + C(y′) 2 + (2Ah + kB + D)x′ + (hB + 2kC + E)y′ + (hBk + Ah2 + Ck2 + hD + Ek + F) = 0

Discriminant of the transformed equation is B2 - 4AC.


Correct answer : (4)
 22.  
Choose the angle to which the axes need to be rotated to remove the x′ y' terms in the transformed equation of 9x2 + 23xy + 3y2 = 0.
a.
π12
b.
π6
c.
π2
d.
π3


Solution:

9x2 + 23xy + 3y2 = 0
[Original equation.]

A = 9 , B = 23 and C = 3
[Compare with Ax2 + Bxy + Cy2 = 0.]

Let α be the angle of rotation needed to eliminate the x' y' term in the transformed equation of the given equation, then cot 2α = A - C / B

α = 1 / 2 Cot-1(A - C / B)
[Solve for α.]

= 1 / 2 Cot-1(9 - 323)
[Substitute the values of A, B, and C.]

= 1 / 2 Cot-1(3)

α = π12

The axes need to be rotated by an angle of π12 to remove the x'y' - term in the transformed equation.


Correct answer : (1)
 23.  
Choose the point to which the axes may be translated so as to remove the first degree terms in the transformed equation of Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.
a.
(14AC-B², 14AC-B²)
b.
(BE - 2CD4AC-B², BD-2AE4AC-B²)
c.
(2CD4AC, BD4AC)
d.
None of the above


Solution:

The point to which the axes may be translated so as to remove the first degree terms in the transformed equation of Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 is (BE-2CD4AC-B², BD-2AE4AC-B²).


Correct answer : (2)
 24.  
Choose the point to which the axes may be translated so as to remove the first degree terms in the transformed equation of 6x2 + 7xy - 8y2 + 5x + 4y + 40 = 0.
a.
(- 108 241, 13 241)
b.
(0, 0 )
c.
(108 241, 0 )
d.
(241 , 0 )


Solution:

6x2 + 7xy - 8y2 + 5x + 4y + 40 = 0
[Original equation.]

A = 6, B = 7, C = - 8, D = 5, E = 4, F = 40
[Compare with Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.]

The point to which axes may be translated is (BE - 2CD4AC-B2, BD - 2AE4AC-B2)

= ( 7(4) - 2(- 8)(5)4(6)(- 8)-(7)2, (7)(5) - 2(6)(4)4(6)(- 8)-(7)2)

= (- 108 / 241, 13 / 241)


Correct answer : (1)
 25.  
Choose the point to which the axes may be translated so as to remove the first degree terms in the transformed equation of xy + 7x - 8y - 41 = 0.
a.
(- 8, 7 )
b.
(8, - 7 )
c.
(- 8, - 7 )
d.
(8, 7 )


Solution:

xy + 7x - 8y - 41 = 0
[Original equation.]

A = 0, B = 1, C = 0, D = 7, E = - 8, F = - 41
[Compare with A x2 + Bxy + C y2 + Dx + Ey + F = 0.]

The point to which axes may be translated is (BE - 2CD4AC-B2, BD - 2AE4AC-B2).

= ( (1)(- 8) - 2(0)(7)4(0)(0)-(1)2, (1)(7) - 2(0)(- 8)4(0)(0)-(1)2 )

= (8, - 7)


Correct answer : (2)

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