﻿ Translation and Rotation of Axes Worksheet | Problems & Solutions

# 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).

2.
Find the discriminant of the equation 9$\mathrm{xy}$ - 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

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.

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'.

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.

6.
Choose the condition when the second degree equation A$x$2 + B$\mathrm{xy}$ + C$y$2 + D$x$ + E$y$ + 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 .

7.
Choose the coordinates of a point P in a plane when the axes are rotated through an angle $\alpha$, where ($x$, $y$) and ($x$′, $y$′) are coordinates of original and rotated axes.
 a. $x$′ = $x$cos $\alpha$; $y$ = - $x$sin $\alpha$ b. $x$′ = $x$cos $\alpha$ + $y$sin $\alpha$; $y$′ = - $x$sin $\alpha$ - $y$cos $\alpha$ c. $x$′ = $x$cos $\alpha$ + $y$sin $\alpha$; $y$′ = - $x$sin $\alpha$ + $y$cos $\alpha$ 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.

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 A$x$2 + B$x$$y$ + C$y$2 + D$x$ + E$y$ + F = 0.
 a. Cot 2$\alpha$ = b. Cot 2$\alpha$ = c. Cot 2$\alpha$ = $\frac{AC}{B}$ d. Cot 2$\alpha$ =

#### 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.

9.
Choose the condition when the second degree equation A$x$2 + B$\mathrm{xy}$ + C$y$2 + D$x$ + E$y$ + 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.

Choose the condition when the second degree equation A$x$2 + B$\mathrm{xy}$ + C$y$2 + D$x$ + E$y$ + $f$ = 0 represents a hyperbola.