Translation and Rotation of Axes Worksheet

**Page 2**

11.

Identitfy the type of conic represented by the equation 6$x$^{2} - 4$x$$y$ + 7$y$^{2} - 40$x$ + 60$y$ - 47 = 0.

a. | a circle | ||

b. | an ellipse | ||

c. | a parabola | ||

d. | a hyperbola |

A = 6, B = - 4, and C = 7

[Compare with A

Discriminant is B

= - 152 < 0

So, the equation represents an ellipse.

[B

Correct answer : (2)

12.

Identify the type of conic represented by the equation 6$x$^{2} - 7$y$^{2} - 2$y$ - 34 = 0.

a. | a parabola | ||

b. | an ellipse | ||

c. | a circle | ||

d. | a hyperbola |

A = 6, B = 0, and C = - 7

[Compare with A

Discriminant is B

= 168 > 0

So, the given equation represents a hyperbola.

[B

Correct answer : (4)

13.

Choose the equation in standard form for the given conic. [Given $a$ = 4, $b$ = 3.]

a. | $x$ ^{2} + $y$^{2}Ã‚Â = 144 | ||

b. | 9$x$ ^{2} + 16$y$^{2} = 144 | ||

c. | 16$x$ ^{2}Ã‚Â + 9$y$^{2}Ã‚Â = 1 | ||

d. | $x$ ^{2}Ã‚Â + 16$y$^{2} = 1 |

[From the figure.]

[Write the standard form of an ellipse.]

So, the equation of the conic shown is

[Substitute the values of

Correct answer : (2)

14.

Choose the co-ordinates of P(5, - 2) in the rotated system, if the axes are rotated through an angle $\frac{\pi}{3}$.

a. | Does not exist | ||

b. | $x$′ = $\frac{5+2\sqrt{3}}{2}$ | ||

c. | $x$′ = $\frac{5-2\sqrt{3}}{2}$, $y$′ = - $\frac{(5\sqrt{3}+2)}{2}$ | ||

d. | $x$′ = $\frac{5+2\sqrt{3}}{2}$, $y$′ = - $\frac{(5\sqrt{3}+2)}{2}$ |

[Substitute

So, the co-ordinates of P in the rotated system are (

Correct answer : (3)

15.

Identify the transformed equation of $x$² + 2$\sqrt{3}$$x$$y$ - $y$² = 2$a$², when the axes are rotated through an angle $\frac{\pi}{6}$.

a. | $x$′² + $y$′² = $a$² | ||

b. | $x$′² + $y$′² = - $a$² | ||

c. | $x$′² - $y$′² = - $a$² | ||

d. | $x$′² - $y$′² = $a$² |

[Substitute

[Substitute

[Original equation.]

(

[Simplify.]

When the axes are rotated through an angle

Correct answer : (4)

16.

Find the approximate angle of rotation needed to eliminate the cross product (or) $x$' $y$' term in the transformed equation of 4$x$^{2} + 2$x$$y$ + 2$y$^{2} - 47 = 0.

a. | $\alpha $ = cot ^{-1} (1 ) | ||

b. | $\alpha $ = $\frac{1}{2}$ tan ^{-1} (1 ) | ||

c. | $\alpha $ = $\frac{1}{2}$ cot ^{-1} (1 ) | ||

d. | $\alpha $ = tan ^{-1} (1 ) |

A = 4, B = 2, and C = 2

[Compare with A

Let

2

[Solve for 2

2

[Substitute the values of A, B, and C.]

[Solve for

Correct answer : (3)

17.

Choose the conic represented by the equation 8$y$^{2} + 48$y$ - 8$x$^{2} + 32$x$ = 24.

a. | an ellipse | ||

b. | a parabola | ||

c. | a circle | ||

d. | a hyperbola |

[Original equation.]

8(

[Group the terms and complete the squares.]

[Divide on both sides by 64.]

The translated equation of the original equation becomes

[Translate the axes using

So, the conic represented by the original equation is a hyperbola.

Correct answer : (4)

18.

Write an equation in standard form for the conic shown.

a. | 9$x$ ^{2}Ã‚Â - 16$y$^{2} = - 144 | ||

b. | 9$x$ ^{2}Ã‚Â - 16$y$^{2} = 144 | ||

c. | 9$x$ ^{2}Ã‚Â + 16$y$^{2} = 144 | ||

d. | 16$x$ ^{2}Ã‚Â - 9$y$^{2} = 144 |

[From the figure,

16

So, the equation of the conic shown is

Correct answer : (4)

19.

Which of the following equations represents the standard form of the conic 16$x$^{2} - $y$^{2} - 32$x$ - 6$y$ - 57 = 0? Translate the axes so that the origin is at the center.

a. | $\frac{{(x\prime )}^{2}}{{2}^{2}}$ - $\frac{{(y\prime )}^{2}}{{8}^{2}}$ = 1 | ||

b. | $\frac{{(x\prime )}^{2}}{{8}^{2}}$ - $\frac{{(y\prime )}^{2}}{{2}^{2}}$ = 1 | ||

c. | $\frac{{(x\prime )}^{2}}{{8}^{2}}$ + $\frac{{(y\prime )}^{2}}{{2}^{2}}$ = 1 | ||

d. | None of the above |

[Original equation of the conic.]

16(

[Group the terms.]

16(

[Complete the squares.]

[Divide both sides by 64.]

The transformed equation of the original equation is

[Translate the conic using

Correct answer : (1)

20.

Which of the following is true about the conic A$x$^{2} + C$y$^{2} + D$x$ + E$y$ + F = 0?

a. | The focal axis of the given equation is not aligned with one of the co-ordinate axes. | ||

b. | It represents an ellipse. | ||

c. | It represents a line. | ||

d. | The focal axis of the given equation is aligned with one of the co-ordinate axes. |

[Original equation of the conic.]

It is clearly known from the equation that the cross product term (

Hence, the focal axis of the given conic is aligned with one of the co-ordinate axes.

Correct answer : (4)