Deck 11: Conic Sections

Question 1

Find the focus, directrix, and focal diameter of the parabola , and sketch its graph.

Answer 1

.Focus: , the directrix is , and the focal diameter is .

Question 2

Find an equation for a parabola with vertex at the origin, and with directrix .

Answer 2

, and the parabola is of the form , so its equation is .

Question 3

Find an equation of the parabola whose graph is shown.

Answer 3

Question 4

Find an equation of the parabola whose graph is shown.

Question 5

Find the focus, directrix, and focal diameter of the parabola , and sketch its graph.

Question 6

Find an equation for a parabola with vertex at the origin, and with focus .

Question 7

Find an equation of a parabola with vertex at the origin, focus on the negative - axis, 7 units away from the directrix.

Question 8

Find an equation of a parabola with horizontal axis and vertex at the origin that passes through the point .

Question 9

Find an equation of the parabola whose graph is shown.

Question 10

Find the focus, directrix, and focal diameter of the parabola , and sketch its graph.

Question 11

A cable is suspended between two utility poles (the shape of the suspended cable is parabolic). If the poles are ft apart, and the lowest point of the cable is ft. below the top of the poles, find the parabolic equation of the suspended cable, placing the origin of the coordinate system at the vertex.

Question 12

Find the focus, directrix, and focal diameter of the parabola , and sketch its graph.

Question 13

Find the focus, directrix, and focal diameter of the parabola , and sketch its graph.

Question 14

Find the focus, directrix, and focal diameter of the parabola , and sketch its graph.

Question 15

Find an equation of the parabola whose graph is shown.

Question 16

Find the focus, directrix, and focal diameter of the parabola , and sketch its graph.

Question 17

Find the focus, directrix, and focal diameter of the parabola , and sketch its graph.

Question 18

Find an equation for a parabola with vertex at the origin, and with directrix .

Question 19

Find an equation of the parabola whose graph is shown.

Question 20

Find an equation for a parabola with vertex at the origin, and with directrix .

Question 21

Find an equation for the ellipse with eccentricity , and foci .

Question 22

Find the vertices, foci, and eccentricity of the ellipse given by . Determine the lengths of the major and minor axes, and sketch the graph.

Question 23

Find the vertices, foci, and eccentricity of the ellipse given by . Determine the lengths of the major and minor axes, and sketch the graph.

Question 24

By placing the origin at the center of Mercury's orbit and the Sun on the - axis at one of the foci, find the equation of Mercury's orbit given that the length of its major axis is m and the elliptical orbit has eccentricity .

Question 25

Find the vertices, foci, and asymptotes of the hyperbola , and sketch its graph.

Question 26

Find an equation for the ellipse whose graph is shown.

Question 27

Find the vertices, foci, and asymptotes of the hyperbola , and sketch its graph.

Question 28

Find the vertices, foci, and eccentricity of the ellipse given by . Determine the lengths of the major and minor axes, and sketch the graph.

Question 29

Find an equation for the ellipse whose foci are , and whose vertices are .

Question 30

Find the vertices, foci, and eccentricity of the ellipse given by . Determine the lengths of the major and minor axes, and sketch the graph.

Question 31

Find an equation for the ellipse with foci , and major axis of length .

Question 32

Find the vertices, foci, and eccentricity of the ellipse given by . Determine the lengths of the major and minor axes, and sketch the graph.

Question 33

Find the vertices, foci, and eccentricity of the ellipse given by . Determine the lengths of the major and minor axes, and sketch the graph.

Question 34

Find an equation for the ellipse whose graph is shown.

Question 35

Find an equation for the ellipse with major axis of length , minor axis of length , foci on the -axis, and centered at the origin.

Question 36

Find an equation for the ellipse with foci , and major axis of length .

Question 37

Find an equation for the ellipse whose graph is shown.

Question 38

Find an equation for the ellipse with endpoints of the minor axis at , and a distance of between the foci.

Question 39

Find an equation for the ellipse with endpoints of the major axis at , and a distance of between the foci.

Question 40

Find the vertices, foci, and eccentricity of the ellipse given by . Determine the lengths of the major and minor axes, and sketch the graph.

Question 41

Find an equation for the hyperbola that has vertices and passes through the point .

Question 42

Find the center, foci, vertices, and asymptotes of the hyperbola , and sketch its graph.

Question 43

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. Then sketch the graph of the equation. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. If the equation has no graph, explain why.

Question 44

Find an equation for the hyperbola that has foci and passes through the point .

Question 45

Find an equation for the conic whose graph is shown.

Question 46

Find the equation for the hyperbola that has asymptotes and passes through the point .

Question 47

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. Then sketch the graph of the equation. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. If the equation has no graph, explain why.

Question 48

Find an equation for the conic whose graph is shown.

Question 49

Find an equation for the hyperbola whose graph is shown.

Question 50

Find the center, foci, vertices, and asymptotes of the hyperbola , and sketch its graph.

Question 51

Find the vertex, focus, and directrix of the parabola , and sketch its graph.

Question 52

Find the vertex, focus, and directrix of the parabola , and sketch its graph.

Question 53

Find an equation for the conic whose graph is shown.

Question 54

Find an equation for the hyperbola that has vertices and asymptotes .

Question 55

Find the vertices, foci, and asymptotes of the hyperbola , and sketch its graph.

Question 56

Find an equation for the hyperbola that has foci and vertices .

Question 57

Find an equation for the hyperbola that has foci and a transverse axis of length .

Question 58

Find an equation for the hyperbola that has foci and vertices .

Question 59

Find an equation for the hyperbola that has foci and vertices .

Question 60

Find an equation for the hyperbola whose graph is shown.

Question 61

Complete the square to determine whether the equation , represents an ellipse, a parabola, a hyperbola, or a degenerate conic. Then sketch the graph of the equation. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. If the equation has no graph, explain why.

Question 62

Determine the - coordinates of if the axes are rotated through an angle .

Question 63

Determine the equation of the conic in - coordinates if the axes are rotated through an angle .

Question 64

Determine the equation of the conic in - coordinates if the axes are rotated through an angle .

Question 65

Use the discriminant to determine if the graph of the equation, is a parabola, an ellipse or a hyperbola, then use a rotation of axes to eliminate the - term, and sketch the graph.

Question 66

A cannon fires a cannonball as shown in the figure. The path of the cannonball is a parabola with vertex at the highest point of the path. If the cannonball lands 1600 ft from the cannon and the highest point it reaches is 2400 ft above the ground, find an equation for the path of the cannonball. Place the origin at the location of the cannon.

Question 67

Use the discriminant to determine if the graph of the equation, is a parabola, an ellipse or a hyperbola, then use a rotation of axes to eliminate the - term, and sketch the graph.

Question 68

Complete the square to determine whether the equation represents an ellipse, parabola, hyperbola or degenerate conic. Then sketch the graph of the equation. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axis. If it is a parabola, find the vertex, focus and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. If the equation has no graph explain why.

Question 69

Use the discriminant to determine if the graph of the equation is a parabola, an ellipse or a hyperbola, then use a rotation of axes to eliminate the - term, and sketch the graph.

Question 70

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. Then sketch the graph of the equation. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. If the equation has no graph, explain why.

Question 71

Determine the - coordinates of if the axes are rotated through an angle .

Question 72

Use the discriminant to determine if the graph of the equation, is a parabola, an ellipse or a hyperbola, then use a rotation of axes to eliminate the - term, and sketch the graph.

Question 73

Determine the - coordinates of if the axes are rotated through an angle .

Question 74

Determine the equation of the conic in - coordinates if the axes are rotated through an angle .

Question 75

Use the discriminant to determine if the graph of the equation, is a parabola, an ellipse or a hyperbola, then use a rotation of axes to eliminate the - term, and sketch the graph.

Question 76

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. Then sketch the graph of the equation. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. If the equation has no graph, explain why.

Question 77

Use the discriminant to determine if the graph of the equation is a parabola, an ellipse or a hyperbola, and then use a rotation of axes to eliminate the - term, and sketch the graph.

Question 78

Complete the square to determine whether the equation represents an ellipse, parabola, hyperbola or degenerate conic. Then sketch the graph of the equation. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axis. If it is a parabola, find the vertex, focus and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. If the equation has no graph explain why.

Question 79

A cannon fires a cannonball as shown in the figure. The path of the cannonball is a parabola with vertex at the highest point of the path. If the cannonball lands 800 ft from the cannon and the highest point it reaches is 1600 ft above the ground, find an equation for the path of the cannonball. Place the origin at the location of the cannon.

Question 80

Complete the square to determine whether the equation represents an ellipse, parabola, hyperbola or degenerate conic. Then sketch the graph of the equation. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axis. If it is a parabola, find the vertex, focus and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. If the equation has no graph explain why.