Translation and Rotation of Axes Worksheet

Translation and Rotation of Axes Worksheet
  • Page 1
 1.  
Choose the coordinates of P (4, 5) in the translated coordinate system, when the origin is shifted to the point P (- 4, 7) without changing the direction of axes.
a.
(- 8, - 2)
b.
(- 8, 2)
c.
(4, - 2)
d.
(8, - 2)


Solution:

The coordinates of P in the new axes are x′ = x - h and y′ = y - k.

x′ = 4 - (- 4) = 8 and y′ = 5 - 7 = - 2
[Substitute x = 4, y = 5, h = - 4, and k = 7.]

Hence, the translated coordinates of P are (8, - 2).


Correct answer : (4)
 2.  
Find the discriminant of the equation 9xy - 8 = 0.
a.
9
b.
81
c.
85


Solution:

9xy - 8 = 0
[Original equation.]

A = 0, B = 9 and C = 0
[Compare with Ax² + Bxy + Cy² + Dx + Ey + F = 0.]

Discriminant = B² - 4AC

= (9)2 - 4(0)(0)
[Substitute the values of A, B, and C.]

= 81


Correct answer : (3)
 3.  
Identify the curve the equation (x - 2)2 = 5(y - 4)2 repesents.
a.
an ellipse
b.
a circle
c.
a parabola
d.
a hyperbola


Solution:

(x - 2)2 = 5(y - 4)2
[Original equation.]

The transformed equation of the original equation is (x′)2 = 8(y′)2 which represents a hyperbola.
[Translate the axes using h = 2 and k = 4 and use x′ = x - 2, y′ = y - 4.]

So, the original equation represents a hyperbola.


Correct answer : (4)
 4.  
Choose the original coordinates of a point P in a plane when the origin O is shifted to the point O'(h, k) without changing the direction of the axes, where (x, y) represents the original coordinates and (x′, y′) represents the changed one.
a.
x = x′ + h; y = y′ + k
b.
x = x′ + k; y = y′ + h
c.
x = x′ - k; y = y′ - h
d.
None of the above


Solution:

If the origin is shifted from the O(0, 0) to O'(h, k) without changing the direction of axes, then the original coordinates of point P are x = x′ + h and y = y′ + k, where (x′, y′) represents the coordinates of P, based on the translated axes and the corresponding origin O'.


Correct answer : (1)
 5.  
Choose the translated coordinates of a point P in a plane when the origin O is shifted to the point O'(h, k) without changing the direction of the axes, where (x, y) and (x′, y′) are the coordinates of the point P referred to original and new axes.
a.
x′ = x - h; y′ = y - k
b.
x′ = x - h; y′ = y + k
c.
x′ = x + h; y′ = y + k
d.
None of the above


Solution:

If the origin is shifted from O to O'(h, k) without changing the direction of axes, then the coordinates of point P based on the translated axes and the corresponding origin O' are x′ = x - h and y′ = y - k where (x, y) are the coordinates of P based on the original axes and the corresponding origin O.


Correct answer : (1)
 6.  
Choose the condition when the second degree equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 represents a parabola.
a.
B2 + 4AC = 0
b.
B2 - 4AC > 0
c.
B2 - 4AC < 0
d.
B2 - 4AC = 0


Solution:

If B2 - 4AC = 0, then the second degree equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 represents a parabola .


Correct answer : (4)
 7.  
Choose the coordinates of a point P in a plane when the axes are rotated through an angle α, where (x, y) and (x′, y′) are coordinates of original and rotated axes.
a.
x′ = xcos α; y = - xsin α
b.
x′ = xcos α + ysin α; y′ = - xsin α - ycos α
c.
x′ = xcos α + ysin α; y′ = - xsin α + ycos α
d.
None of the above


Solution:

When the axes are rotated through an angle α, then the coordinates of point P in the translated system are x′ = xcos α + ysin α and y′ = - xsin α + ycos α.


Correct answer : (3)
 8.  
Choose the angle to which the axes need to be rotated to remove the cross-product term (x'y' - term) in the translated equation of the original equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.
a.
Cot 2α = A - C B
b.
Cot 2α = A - C D
c.
Cot 2α = AC B
d.
Cot 2α = A - B C


Solution:

The angle to which the axes need to be rotated to remove the cross-product term (x'y' - term) in the translated equation of the original equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 is an acute angle α such that Cot 2α = A - C / B.


Correct answer : (1)
 9.  
Choose the condition when the second degree equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 represents an ellipse.
a.
B2 - 4AC = 0
b.
B2 + 4AC < 0
c.
B2 - 4AC < 0
d.
B2 - 4AC > 0


Solution:

If B2 - 4AC < 0, then the second degree equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 represents an ellipse.


Correct answer : (3)
 10.  
Choose the condition when the second degree equation Ax2 + Bxy + Cy2 + Dx + Ey + f = 0 represents a hyperbola.
a.
B2 - 4AC < 0
b.
B2 - 4AC > 0
c.
B2 - 4AC = 0
d.
B2 + 4AC > 0


Solution:

If B2 - 4AC > 0, then the second degree equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 represents a hyperbola.


Correct answer : (2)

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