Deck 6: Systems and Matrices

Question 1

Match each equation on the left with the appropriate description of its graph on the right.

- $(y+3)^{2}-x=-4$

A) Ellipse; center $(-4,-3)$
B) Parabola; opens right
C) Circle; center (4, -3 $)$ ; radius 6
D) Parabola; opens upward
E) Circle; center $(-4,3)$ ; radius 6
F) Hyperbola; center $(4,-3)$

Answer 1

Parabola; opens right

Question 2

Match each equation on the left with the appropriate description of its graph on the right.

- $9(x-4)^{2}-4(y+3)^{2}=36$

A) Ellipse; center $(-4,-3)$
B) Parabola; opens right
C) Circle; center (4, -3 $)$ ; radius 6
D) Parabola; opens upward
E) Circle; center $(-4,3)$ ; radius 6
F) Hyperbola; center $(4,-3)$

Answer 2

Hyperbola; center $(4,-3)$

Question 3

Match each equation on the left with the appropriate description of its graph on the right.

- $(x-4)^{2}+(y+3)^{2}=36$

A) Ellipse; center $(-4,-3)$
B) Parabola; opens right
C) Circle; center (4, -3 $)$ ; radius 6
D) Parabola; opens upward
E) Circle; center $(-4,3)$ ; radius 6
F) Hyperbola; center $(4,-3)$

Answer 3

Circle; center (4, -3 $)$ ; radius 6

Question 4

Match each equation on the left with the appropriate description of its graph on the right.

- $(x+4)^{2}-y=3$

A) Ellipse; center $(-4,-3)$
B) Parabola; opens right
C) Circle; center (4, -3 $)$ ; radius 6
D) Parabola; opens upward
E) Circle; center $(-4,3)$ ; radius 6
F) Hyperbola; center $(4,-3)$

Question 5

Match each equation on the left with the appropriate description of its graph on the right.

- $9(x+4)^{2}+4(y+3)^{2}=36$

A) Ellipse; center $(-4,-3)$
B) Parabola; opens right
C) Circle; center (4, -3 $)$ ; radius 6
D) Parabola; opens upward
E) Circle; center $(-4,3)$ ; radius 6
F) Hyperbola; center $(4,-3)$

Question 6

Match each equation on the left with the appropriate description of its graph on the right.

- $(x+4)^{2}+(y-3)^{2}=36$

A) Ellipse; center $(-4,-3)$
B) Parabola; opens right
C) Circle; center (4, -3 $)$ ; radius 6
D) Parabola; opens upward
E) Circle; center $(-4,3)$ ; radius 6
F) Hyperbola; center $(4,-3)$

Question 7

Give the coordinates of the focus and the equation of the directrix for the parabola with equation $y=-4 x^{2}$ .

Question 8

Graph $y=2 \sqrt{\frac{x^{2}}{4}-1}$ . Is the graph that of a function? Find the domain and range.

Question 9

The screen shown here gives the graph of $\frac{x^{2}}{16}+\frac{y^{2}}{25}=1$ as generated by a graphing calculator.
What two functions were used to obtain the graph?

Question 10

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable.

- $x=t-1, y=-13 \sqrt{1-\frac{t^{2}}{25}}+3, t$ in $[-5,5]$

Question 11

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable.

- $\frac{(y+2)^{2}}{9}-\frac{(x-1)^{2}}{16}=1$

Question 12

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable.

- $y^{2}-4 x-2 y+13=0$

Question 13

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable.

- $\frac{y^{2}}{9}-\frac{x^{2}}{1}=1$

Question 14

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable.

- $x^{2}+y^{2}-6 x-4 y+13=0$

Question 15

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable.

- $\frac{(x-1)^{2}}{25}+\frac{(y+1)^{2}}{64}=1$

Question 16

Write an equation for each of the following.
(a) The conic with focus at the origin and vertex at $(0,3)$ and $e=1$ .
(b) The conic with center at the origin, focus at $(-2,0)$ and $e=\frac{2}{3}$ .

Question 17

Some comets orbit the sun in elliptical orbits with the sun at one focus. The path of the orbit of Halley's Comet has a major axis of length 35.8 astronomical units (AU) and an eccentricity of .967. (Source: NASA.) Position the center of the ellipse at the origin and determine the equation of the ellipse. What is the minimum distance between Halley's Comet and the sun?

Question 18

Match each equation on the left with the appropriate description of its graph on the right.

- $4(x-6)^{2}-25(y+2)^{2}=100$

A) Circle; center $(6,-2)$ ; radius 5
B) Hyperbola; center $(6,-2)$
C) Parabola; opens left
D) Circle; center $(-6,2)$ ; radius 5
E) Parabola; opens downward
F) Ellipse; center $(6,2)$

Question 19

Match each equation on the left with the appropriate description of its graph on the right.

- $(x+6)^{2}+y=2$

A) Circle; center $(6,-2)$ ; radius 5
B) Hyperbola; center $(6,-2)$
C) Parabola; opens left
D) Circle; center $(-6,2)$ ; radius 5
E) Parabola; opens downward
F) Ellipse; center $(6,2)$

Question 20

Match each equation on the left with the appropriate description of its graph on the right.

- $(x-6)^{2}+(y+2)^{2}=25$

A) Circle; center $(6,-2)$ ; radius 5
B) Hyperbola; center $(6,-2)$
C) Parabola; opens left
D) Circle; center $(-6,2)$ ; radius 5
E) Parabola; opens downward
F) Ellipse; center $(6,2)$

Question 21

Match each equation on the left with the appropriate description of its graph on the right.

- $4(x-6)^{2}+25(y-2)^{2}=100$

A) Circle; center $(6,-2)$ ; radius 5
B) Hyperbola; center $(6,-2)$
C) Parabola; opens left
D) Circle; center $(-6,2)$ ; radius 5
E) Parabola; opens downward
F) Ellipse; center $(6,2)$

Question 22

Match each equation on the left with the appropriate description of its graph on the right.

- $(y+2)^{2}+x=6$

A) Circle; center $(6,-2)$ ; radius 5
B) Hyperbola; center $(6,-2)$
C) Parabola; opens left
D) Circle; center $(-6,2)$ ; radius 5
E) Parabola; opens downward
F) Ellipse; center $(6,2)$

Question 23

Match each equation on the left with the appropriate description of its graph on the right.

- $(x+6)^{2}+(y-2)^{2}=25$

A) Circle; center $(6,-2)$ ; radius 5
B) Hyperbola; center $(6,-2)$
C) Parabola; opens left
D) Circle; center $(-6,2)$ ; radius 5
E) Parabola; opens downward
F) Ellipse; center $(6,2)$

Question 24

Give the coordinates of the focus and the equation of the directrix for the parabola with equation $x=-2 y^{2}$ .

Question 25

Graph $y=2 \sqrt{1-\frac{x^{2}}{4}}$ . Is the graph that of a function? Find the domain and range.

Question 26

The screen shown here gives the graph of $\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$ as generated by a graphing calculator.
What two functions were used to obtain the graph?

Question 27

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable

- $y^{2}+4 x+10 y+13=0$

Question 28

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable

- $x=t+1, y=3 \sqrt{1-\frac{t^{2}}{25}}-2, t$ in $[-5,0]$

Question 29

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable

- $x^{2}+y^{2}-16 x-6 y+69=0$

Question 30

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable

- $\frac{x^{2}}{64}-\frac{y^{2}}{36}=1$

Question 31

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable

- $x^{2}+y^{2}-10 x+4 y+29=0$

Question 32

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable

- $\frac{y^{2}}{25}+\frac{x^{2}}{4}=1$

Question 33

Write an equation for each of the following.
(a) The conic with center at the origin, focus at $(10,0)$ and $e=\frac{5}{6}$ .
(b) The conic centered at the origin with focus at $(0,13)$ and $e=\frac{13}{5}$ .

Question 34

A gallery has the shape of the top half of an ellipse. The cross section is 13 feet high at the center and 40 feet long. Find the equation of the complete ellipse. How far apart are the foci?

Question 35

Match each equation on the left with the appropriate description of its graph on the right.

- $(x+3)^{2}+y=1$

A) Ellipse; center $(3,1)$
B) Hyperbola; center $(3,1)$
C) Circle; center $(3,-1)$ ; radius 4
D) Parabola; opens downward
E) Circle; center $(-3,1)$ ; radius 4
F) Parabola; opens left

Question 36

Match each equation on the left with the appropriate description of its graph on the right.

- $(y-1)^{2}+x=3$

A) Ellipse; center $(3,1)$
B) Hyperbola; center $(3,1)$
C) Circle; center $(3,-1)$ ; radius 4
D) Parabola; opens downward
E) Circle; center $(-3,1)$ ; radius 4
F) Parabola; opens left

Question 37

Match each equation on the left with the appropriate description of its graph on the right.

- $(x-3)^{2}+4(y-1)^{2}=16$

A) Ellipse; center $(3,1)$
B) Hyperbola; center $(3,1)$
C) Circle; center $(3,-1)$ ; radius 4
D) Parabola; opens downward
E) Circle; center $(-3,1)$ ; radius 4
F) Parabola; opens left

Question 38

Match each equation on the left with the appropriate description of its graph on the right.

- $(x+3)^{2}+(y-1)^{2}=16$

A) Ellipse; center $(3,1)$
B) Hyperbola; center $(3,1)$
C) Circle; center $(3,-1)$ ; radius 4
D) Parabola; opens downward
E) Circle; center $(-3,1)$ ; radius 4
F) Parabola; opens left

Question 39

Match each equation on the left with the appropriate description of its graph on the right.

- $(x-3)^{2}+(y+1)^{2}=16$

A) Ellipse; center $(3,1)$
B) Hyperbola; center $(3,1)$
C) Circle; center $(3,-1)$ ; radius 4
D) Parabola; opens downward
E) Circle; center $(-3,1)$ ; radius 4
F) Parabola; opens left

Question 40

Match each equation on the left with the appropriate description of its graph on the right.

- $(x-3)^{2}-4(y-1)^{2}=16$

A) Ellipse; center $(3,1)$
B) Hyperbola; center $(3,1)$
C) Circle; center $(3,-1)$ ; radius 4
D) Parabola; opens downward
E) Circle; center $(-3,1)$ ; radius 4
F) Parabola; opens left

Question 41

Give the coordinates of the focus and the equation of the directrix for the parabola with equation $x=5 y^{2}$ .

Question 42

Graph $y=-3 \sqrt{1-\frac{x^{2}}{16}}$ . Is the graph that of a function? Find the domain and range.

Question 43

The screen shown here gives the graph of $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$ as generated by a graphing calculator.
What two functions were used to obtain the graph?

Question 44

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable.

- $x^{2}+y^{2}-10 x+4 y+13=0$

Question 45

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable.

- $x=t+2, y=-4 \sqrt{1-\frac{t^{2}}{25}}+5, t$ in $[0,5]$

Question 46

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable.

- $x^{2}+y^{2}+6 x-8 y-11=0$

Question 47

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable.

- $x^{2}+y^{2}+14 x-6 y+57=0$

Question 48

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable.

- $\frac{x^{2}}{9}+\frac{y^{2}}{25}=1$

Question 49

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable.

- $x^{2}+y^{2}+4 x+4 y+8=0$

Question 50

Write an equation for each of the following.
(a) The conic with center at the origin, focus at $(3,0)$ and $e=\frac{3}{2}$ .
(b) The conic with focus at the origin, vertex at $(-5,0)$ and $e=1$ .

Question 51

Planets in our solar system orbit the sun in elliptical orbits with the sun at one focus. The path of the orbit of Mercury has a major axis of length 115.8 million miles and an eccentricity of .206. (Source: NASA.) Position the center of the ellipse at the origin and determine the equation of the ellipse. What is the maximum distance between Mercury and the sun?

Question 52

Match each equation on the left with the appropriate description of its graph on the right.

- $(x+2)^{2}+(y-1)^{2}=9$

A) Circle; center $(2,-1)$ ; radius 3
B) Hyperbola; center $(2,1)$
C) Parabola; opens left
D) Circle; center $(-2,1)$ ; radius 3
E) Parabola; opens downward
F) Ellipse; center $(2,1)$

Question 53

Match each equation on the left with the appropriate description of its graph on the right.

- $(y+1)^{2}+x=2$

A) Circle; center $(2,-1)$ ; radius 3
B) Hyperbola; center $(2,1)$
C) Parabola; opens left
D) Circle; center $(-2,1)$ ; radius 3
E) Parabola; opens downward
F) Ellipse; center $(2,1)$

Question 54

Match each equation on the left with the appropriate description of its graph on the right.

- $3(x-2)^{2}+(y-1)^{2}=9$

A) Circle; center $(2,-1)$ ; radius 3
B) Hyperbola; center $(2,1)$
C) Parabola; opens left
D) Circle; center $(-2,1)$ ; radius 3
E) Parabola; opens downward
F) Ellipse; center $(2,1)$

Question 55

Match each equation on the left with the appropriate description of its graph on the right.

- $3(x-2)^{2}-(y-1)^{2}=9$

A) Circle; center $(2,-1)$ ; radius 3
B) Hyperbola; center $(2,1)$
C) Parabola; opens left
D) Circle; center $(-2,1)$ ; radius 3
E) Parabola; opens downward
F) Ellipse; center $(2,1)$

Question 56

Match each equation on the left with the appropriate description of its graph on the right.

- $(x-2)^{2}+(y+1)^{2}=9$

A) Circle; center $(2,-1)$ ; radius 3
B) Hyperbola; center $(2,1)$
C) Parabola; opens left
D) Circle; center $(-2,1)$ ; radius 3
E) Parabola; opens downward
F) Ellipse; center $(2,1)$

Question 57

Match each equation on the left with the appropriate description of its graph on the right.

- $(x-2)^{2}+y=1$

A) Circle; center $(2,-1)$ ; radius 3
B) Hyperbola; center $(2,1)$
C) Parabola; opens left
D) Circle; center $(-2,1)$ ; radius 3
E) Parabola; opens downward
F) Ellipse; center $(2,1)$

Question 58

Give the coordinates of the focus and the equation of the directrix for the parabola with equation $y=3 x^{2}$ .

Question 59

Graph $y=-\sqrt{\frac{x^{2}}{25}-1}$ . Is the graph that of a function? Find the domain and range.

Question 60

The screen shown here gives the graph of $\frac{x^{2}}{9}+\frac{y^{2}}{36}=1$ as generated by a graphing calculator.
What two functions were used to obtain the graph?

Question 61

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable.

- $\frac{(x-1)^{2}}{9}-\frac{(y+4)^{2}}{16}=1$

Question 62

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable.

- $x^{2}-4 y-10 x+1=0$

Question 63

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable.

- $x=t-3, y=10 \sqrt{1-\frac{t^{2}}{36}}+2, t$ in $[-6,6]$

Question 64

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable.

- $x^{2}+4 y+12 x+16=0$

Question 65

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable.

- $x^{2}+y^{2}-8 x+2 y+17=0$

Question 66

Graph the relation. Identify the graph and give the the radius, center, vertices, and foci as applicable.

- $\frac{(x+2)^{2}}{36}+\frac{(y-1)^{2}}{16}=1$

Question 67

Write an equation for each of the following.
(a) The conic with center at the origin, focus at $(10,0)$ and $e=\frac{5}{3}$ .
(b) The conic centered at the origin with vertical major axis of length 6 and $e=\frac{1}{3}$ .

Question 68

A comet following a parabolic path with the Earth at the focus came within 8 million miles of the Earth. Position the Earth at the origin and the perihelion (closest) point of the orbit on the positive horizontal axis and determine the equation of the parabola. How far is the comet from Earth when it crosses the vertical axis?