Translating Conic Sections Worksheet

Translating Conic Sections Worksheet
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1.
What is the minor axis of the ellipse 100$x$2 + 36$y$2 = 3600?
 a. a horizontal line b. a line with slope - 1 c. a line with slope1 d. a vertical line

Solution:

100x2 + 36y2 = 3600
[Equation of the ellipse.]

x236 + y2100 = 1
[Divide throughout by 3600.]

The minor axis of the ellipse is horizontal.
[Compare with x2b2 + y2a2 = 1.]

2.
Find the vertices of the hyperbola 9$x$2 - 4$y$2 = 36.
 a. (± 5, 0) b. (± 2, 0) c. (0, ± 2) d. (0, ± 3)

Solution:

9x2 - 4y2 = 36
[Equation of hyperbola.]

x24 - y29 = 1
[Divide throughout by 36.]

a2 = 4 a = 2
b2 = 9 b = 3
[Compare with x2a2 - y2b2 = 1.]

The vertices of the hyperbola are (± 2, 0).
[Use (± a, 0).]

3.
Find the vertices of the ellipse 81$x$2 + 64$y$2 = 5184.
 a. (- 8, 0), (8, 0) b. (9, 0), (- 9, 0) c. (0, 8), (0, - 8) d. (0, 9), (0, - 9)

Solution:

81x2 + 64y2 = 5184
[Equation of the ellipse.]

x264 + y281 = 1
[Divide throughout by 5184.]

b2 = 64 b = 8
a2 = 81 a = 9
[Compare with x2b2 + y2a2 = 1.]

The vertices of the ellipse are (0, 9) and (0, - 9).
[Use (0, ± a).]

4.
What are the co-vertices of the ellipse 81$x$2 + 100$y$2 = 8100?
 a. (0, - 9), (0, 9) b. (10, 0), (- 10, 0) c. (0, 10), (0, - 10) d. (9, 0), (- 9, 0)

Solution:

81x2 + 100y2 = 8100
[Equation of the ellipse.]

x2100 + y281 = 1
[Divide throughout by 8100.]

a2 = 100 a = 10
b2 = 81 b = 9
[Compare with x2a2 + y2b2 = 1.]

The co-vertices of the ellipse are (0, - 9) and (0, 9).
[Use (0, ± b).]

5.
Choose the conic section represented by the equation.
$x$2 + $y$2 - 16$x$ + 16$y$ + 64 = 0
 a. a hyperbola b. a parabola c. a circle d. an ellipse

Solution:

x2 + y2 - 16x + 16y + 64 = 0
[Equation of conic section.]

x2 - 16x + y2 + 16y = - 64
[Group the x and y - terms.]

(x2 - 16x + ?) + (y2 + 16y + ?) = - 64

(x2 - 16x + 64) + (y2 + 16y + 64) = - 64 + 64 + 64

(x - 8)2 + (y + 8)2 = (8)2
[Complete the squares.]

The equation represents a circle with center (8, - 8) and radius 8.
[Compare with (x - h)2 + (y - k)2 = r2.]

6.
Find the center of the conic section represented by the equation.
$x$2 + $y$2 - 2$x$ + 2$y$ + 1 = 0
 a. (- 1, 1) b. (1, - 1) c. (1, 1) d. (1, 0)

Solution:

x2 + y2 - 2x + 2y + 1 = 0
[Equation of conic section.]

x2 - 2x + y2 + 2y = - 1
[Group x-terms and y - terms.]

(x2 - 2x + ?) + (y2 + 2y + ?) = - 1

(x2 - 2x + 1) + (y2 + 2y + 1) = - 1 + 1 + 1

(x - 1)2 + (y + 1)2 = (1)2
[Complete the squares.]

The equation represents a circle with center (1, - 1) and radius 1.
[Compare with (x - h)2 + (y - k)2 = r2.]

7.
Find the standard form of the equation.
$x$2 + $y$2 - 4$x$ + 2$y$ + 1 = 0
 a. ($x$ - 2)2 + ($y$ + 1)2 = 22 b. ($x$ + 2)2 + ($y$ + 1)2 = 22 c. ($x$ + 2)2 + ($y$ - 1)2 = 22 d. ($x$ - 2)2 + ($y$ - 1)2 = 22

Solution:

x2 + y2 - 4x + 2y + 1 = 0
[Equation of conic section.]

x2 - 4x + y2 + 2y = - 1
[Group x and y - terms.]

(x2 - 4x + ?) + (y2 + 2y + ?) = - 1

(x2 - 4x + 4) + (y2 + 2y + 1) = - 1 + 4 + 1

(x - 2)2 + (y + 1)2 = (2)2
[Complete the squares.]

So, the standard form of the given equation is (x - 2)2 + (y + 1)2 = (2)2, which represents a circle with center (2, - 1) and radius 2.

8.
What conic section is represented by the equation
9$x$2 + 4$y$2 - 18$x$ - 8$y$ - 23 = 0?
 a. a circle b. a parabola c. a hyperbola d. an ellipse

Solution:

9x2 + 4y2 - 18x - 8y - 23 = 0
[Equation of conic section.]

9x2 - 18x + 4y2 - 8y = 23
[Group x - terms and y - terms.]

9(x2 - 2x) + 4(y2 - 2y) = 23

9(x2 - 2x + ?) + 4(y2 - 2y + ?) = 23

9(x2 - 2x + 1) + 4(y2 - 2y + 1) = 23 + 9 + 4

9(x - 1)2 + 4(y - 1)2 = 36
[Complete the squares.]

(x - 1)24 + (y - 1)29 = 1
[Divide throughout by 36.]

The equation represents an ellipse whose center is (1, 1) and major axis is vertical.
[Compare with (x - h)2a2 + (y - k)2b2 = 1.]

9.
Find the center of the conic section represented by the equation.
$x$2 + 9$y$2 + 2$x$ - 18$y$ + 1 = 0
 a. (1, - 1) b. (- 1, 1) c. (9, 1) d. (3, 1)

Solution:

x2 + 9y2 + 2x - 18y + 1 = 0
[Equation of conic section.]

x2 + 2x + 9y2 - 18y = - 1
[Group x and y - terms.]

(x2 + 2x) + 9(y2 - 2y) = - 1

(x2 + 2x + ?) + 9(y2 - 2y + ?) = - 1

(x2 + 2x + 1) + 9(y2 - 2y + 1) = - 1 + 1 + 9

(x + 1)2 + 9(y - 1)2 = 9
[Complete the squares.]

(x + 1)29 + (y - 1)21 = 1
[Divide throughout by 9.]

The equation represents an ellipse whose center is (- 1, 1) and major axis is horizontal.
[Compare with (x - h)2a2 + (y - k)2b2 = 1.]

10.
Find the major axis of the ellipse represented by the equation $x$2 + 4$y$2 + 2$x$ + 16$y$ + 13 = 0.
 a. horizontal major axis b. a line with slope 1 c. a line with slope - 1 d. vertical axis

Solution:

x2 + 4y2 + 2x + 16y + 13 = 0
[Equation of ellipse.]

(x2 + 2x) + (4y2 + 16y) = - 13
[Group x and y - terms.]

(x2 + 2x) + 4(y2 + 4y) = - 13

(x2 + 2x + ?) + 4(y2 + 4y + ?) = - 13

(x2 + 2x + 1) + 4(y2 + 4y + 4) = - 13 + 1 + 16

(x + 1)2 + 4 (y + 2)2 = 4
[Complete the squares.]

(x + 1)24 + (y + 2)21 = 1
[Divide throughout by 4.]

The equation represents an ellipse with center (- 1, - 2) and a horizontal major axis.
[Compare with (x - h)2a2 + (y - k)2b2 = 1.]