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Rotation of Conics Worksheet

Rotation of Conics Worksheet
  • Page 1
 1.  
Choose the coordinates of a point (- 4, - 4) when the axes are turned through an angle 90°.
a.
(4, 4)
b.
(4, - 4)
c.
(- 4, 4)
d.
(- 4, - 4)


Solution:

When the axes are turned through an angle θ, then the new coordinates in terms of original coordinates are x′ = x cos θ + y sin θ and y′ = -x sin θ + y cos θ.
[Formula.]

x′ = (- 4) cos 90° + (- 4) sin 90°
[Substitute the values.]

= (- 4)(0) + (-4)(1) = - 4

y′ = -(-4) sin 90° + (-4) cos 90°
[Substitute the values.]

= (4)(1) + (-4)(0) = 4

The new coordinates are (- 4, 4).


Correct answer : (3)
 2.  
Choose the original coordinates of a point P in terms of new coordinates, when the axes are rotated through an angle θ without changing the origin, where (x, y) and (x′, y′) be the original and new coordinates.
a.
x = x′ cos θ and y = y′ sin θ
b.
x′ = x cos θ and y′ = y sin θ
c.
x = x′ cos θ - y′ sin θ and y = x′ sin θ + y′ cos θ
d.
None of the above


Solution:

When the axes are rotated through an angle θ, then the original coordinates in terms of new coordinates will be x = x′ cos θ - y′ sin θ and y = x′ sin θ + y′ cos θ.


Correct answer : (3)
 3.  
The transformed equation of a conic after the axis are rotated through some angle is given by x2 + 483xy + 2y2 + (5 + 23)x + (53 - 2)y + 48 = 0. The conic is ________.
a.
ellipse
b.
circle
c.
hyperbola
d.
parabola


Solution:

x2 + 483xy + 2y2 + (5 + 23)x + (53- 2)y + 48 = 0 is the given transformed equation of a conic.

Comparing the given equation with Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, we get A = 1, B = 483, and C = 5.

Discriminant = B2 - 4AC = = (483)2 - 4(1) (2) = 6904

Since B2 - 4AC > 0, the given conic equation represents a hyperbola.


Correct answer : (3)
 4.  
Choose the new coordinates of a point P in terms of original coordinates, when the axes are rotated through an angle θ without changing the origin, where (x, y) and (x′, y′) be the original and new coordinates.
a.
x′ = x cos θ + y sin θ and y′ = -x sin θ + y cos θ
b.
x = x′ cos θ and y = y′ sin θ
c.
x′ = x cos θ and y′ = y sin θ
d.
None of the above


Solution:

When the axes are rotated through an angle θ then the new coordinates of a point P in terms of original coordinates are x′ = x cos θ + y sin θ and y′ = -x sin θ + y cos θ.


Correct answer : (1)
 5.  
Choose the angle to which the axes are to be rotated so that the x' y' term in the translated equation of Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 will not be present.
a.
θ = 1 2Cot-1[A-C B]
b.
θ = cot-1[A-C B]
c.
θ = 1 2cot-1[A-B C]
d.
θ = 1 2cot-1[A B]


Solution:

The x' y' - term in the transformed equation will be removed when the axes are turned through an angle θ = 1 / 2 Cot-1[A-C / B].


Correct answer : (1)
 6.  
Choose the coordinates of the point P (7, 0) in original system, when the original system is rotated through an angle π4.
a.
(72, 0)
b.
(0, 72)
c.
(72, 72)
d.
(12, 12)


Solution:

When the axes are rotated through an angle θ, then the original coordinates in terms of new coordinates will be x = x′ cos θ - y′ sin θ and y = x′ sin θ + y′ cos θ.

x = x′ cos θ - y′ sin θ

= (7) cos (π4) - (0) sin (π4)
[Substitute the values.]

= 72 - 0 = 72

y = x′ sin θ + y′ cos θ

= (7) sin (π4) + (0) cos(π4)
[Substitute the values.]

= 72 + 0 = 72

Hence, the coordinates of the point P in the original system are (72, 72).


Correct answer : (3)
 7.  
Choose the coordinates of the point P (0, 10) in the original system, when the original system is rotated through an angle π3.
a.
[ 5 3, 5 ]
b.
[ 3, 5 ]
c.
[- 5 3, 5 ]
d.
[ 5 3, - 5 ]


Solution:

When the axes are rotated through an angle θ, then the original coordinates in terms of new coordinates will be x = x′ cos θ - y′ sin θ and y = x′ sin θ + y′ cos θ.
[Formula.]

x = x′ cos θ - y′ sin θ

= (0) cos (π3) - (10) sin (π3)
[Substitute the values.]

= - 5 3

y = x′ sin θ + y′ cos θ

= (0) sin (π3) + (10) cos (π3)
[Substitute the values.]

= 5

Hence, the coordinates of point P in original system is [- 5 3, 5 ].


Correct answer : (3)
 8.  
Choose the conic, which is represented by the equation x2 - xy + y2 = 2 when it is rotated through a suitable angle.
a.
Ellipse
b.
Hyperbola
c.
Parabola
d.
None of the above


Solution:

x2 - xy + y2 = 2

Comparing the given equation with Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, we get A = 1, B = -1, C = 1, D = 0, E = 0, and F = -2.

So, θ = 1 / 2 Cot-1[A-C / B] θ = 1 / 2 Cot-1[1-1-1]

= 12 Cot-1(0) = π4

The second degree equation obtained when we apply rotation to Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 is A′x2 + B′xy + C′y2 + D′x + E′y + F′ = 0 where the values of coefficients are calculated as follows.
[Formula.]

A′ = (1)( 1 / 2) + (-1)( 1 / 2) + (1)( 1 / 2) = 1 / 2
[Substitute in A′ = A cos² θ + B cos θ sin θ + C sin² θ.]

B′ = (-1) (0) + (0) (1) = 0
[Substitute in B′ = B cos 2θ + (C - A) sin 2θ.]

C′ = (1) (1 / 2) - (-1) (12) (12) + (1) (1 / 2) = 3 / 2
[Substitute in C′ = C cos² θ - B cos θ sin θ + A sin² θ.]

D′ = (0) 12 + (0) 12 = 0
[Substitute in D′ = D cos θ + E sin θ.]

E′ = (0) 12 - (0) 12 = 0
[Substitute in E′ = E cos θ - D sin θ.]

F′ = -2
[Substitute in F′ = F.]

Hence, the equation in rotated system is (1 / 2)x2 + (0)xy + (3 / 2)y2 + (0)x + (0)y - 2 = 0
[Substitute the values.]

x2 + 3y2 = 4 or x24 + y2(43) = 1

Clearly, the above equation represents an ellipse.


Correct answer : (1)
 9.  
Choose the new coordinates of the point (- 10, 9) when the axes are rotated through an angle 30°.
a.
[- 103+92, 10+932]
b.
[103-92, 10+932]
c.
[- 103+92, 10- 932]
d.
[ 103+92, 10+932]


Solution:

When the axes are rotated through an angle θ then the new coordinates in terms of original coordinates are x′ = x cos θ + y sin θ and y′ = -x sin θ + y cos θ.
[Formula.]

x′ = (- 10) cos 30° + (9) sin 30°
[Substitute the values.]

= (- 10)32 + 9 / 2

= - 103+92

y′ = -(- 10) sin 30° + (9) cos 30°
[Substitute the values.]

= 10 / 2 + 932

= 10+932

New coordinates = [- 103+92, 10+932]


Correct answer : (1)
 10.  
Choose the angle to which the axes are to be rotated to remove the x' y' - term in the transformed equation of 9x2 + 3xy + 9y2 - 44 = 0.
a.
π6
b.
π4
c.
π
d.
π2


Solution:

The angle to which the axes are to be rotated to remove the x' y' - term in the transformed equation of Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 is θ = 1 / 2 Cot-1[A-C / B].

A = 9, B = 3, and C = 9
[Compare with standard second degree equation.]

θ = 1 / 2Cot-1[9-93]
[Substitute the values.]

= 1 / 2Cot-1(0) = 1 / 2(π2)

= π4


Correct answer : (2)

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