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) |

[Substitute

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

Correct answer : (4)

2.

Find the discriminant of the equation 9$\mathrm{xy}$ - 8 = 0.

a. | 9 | ||

b. | 81 | ||

c. | 85 |

[Original equation.]

A = 0, B = 9 and C = 0

[Compare with A

Discriminant = B² - 4AC

= (9)

[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 |

[Original equation.]

The transformed equation of the original equation is (

[Translate the axes using

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 |

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 |

Correct answer : (1)

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. | B ^{2} + 4AC = 0 | ||

b. | B ^{2} - 4AC > 0 | ||

c. | B ^{2} - 4AC < 0 | ||

d. | B ^{2} - 4AC = 0 |

Correct answer : (4)

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 |

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 A$x$^{2} + B$x$$y$ + C$y$^{2} + D$x$ + E$y$ + F = 0.

a. | Cot 2$\alpha $ = $\frac{\mathrm{A}-\mathrm{C}}{\mathrm{B}}$ | ||

b. | Cot 2$\alpha $ = $\frac{\mathrm{A}-\mathrm{C}}{\mathrm{D}}$ | ||

c. | Cot 2$\alpha $ = $\frac{\mathrm{A}\mathrm{C}}{\mathrm{B}}$ | ||

d. | Cot 2$\alpha $ = $\frac{\mathrm{A}-\mathrm{B}}{\mathrm{C}}$ |

Correct answer : (1)

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. | B ^{2} - 4AC = 0 | ||

b. | B ^{2} + 4AC < 0 | ||

c. | B ^{2} - 4AC < 0 | ||

d. | B ^{2} - 4AC > 0 |

Correct answer : (3)

10.

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.

a. | B ^{2} - 4AC < 0 | ||

b. | B ^{2} - 4AC > 0 | ||

c. | B ^{2} - 4AC = 0 | ||

d. | B ^{2} + 4AC > 0 |

Correct answer : (2)