17 Conic Sections

Question 1

What do Parabols, Hyperbolas, Circles and Ellipses have in common?

Answer 1

They are all formed by quadratic equations.

Question 2

How is a parabola formed?

Answer 2

By a directrix, a line in the plane, and a focus, a point not on the directrix.

A parabola is the set of all points equidistant from the directrix and the focus.

Question 3

What 2 types of parabolas are there?

Answer 3

x² parabolas (face up or down) and

y² parabolas (face right or left - not functions).

Question 4

What is a hyperbola?

Answer 4

A hyperbola has 2 pieces that are mirror images. It is formed by the intersection of a plane with a double cone.

Question 5

When do hyperbolas arise in nature/observation?

Answer 5

(1) the curve y(x) = 1/x
(2) the path of a spacecraft using gravity-assisted swing-by of a planet
(3) the path of a comet travelling to fast to ever return to the solar system

Question 6

Describe the shape of a hyperbola.

Answer 6

Each branch has 2 arms which become straighter further out from the center.

Question 7

What does this create?

Answer 7

2 diagonally-opposite arms which serve as asymptotes.

Where they intersect is the center of symmetry for the whole hyperbola.

With y = 1/x, the asymptotes are the x- and y- axes.

Question 8

What are the components of a Hyperbola?

Answer 8

It has 2 foci, a center, 2 asymptotes, semi-major axis, linear eccentricity and eccentricity

Question 9

How is a Hyperbola formed?

Answer 9

It is the locus of points such that any point, P, such that the absolute value of the difference of the distance from the 2 foci is constant.

That constant is 2 times the semi-major axis or the distnce between the vertex of each branch.

Question 10

What are the axis of symmetry, the vertices, the constant and the asymptotes of a Hyperbola?

Answer 10

The axis of symmetry is the line that goes through the 2 foci.

The vertices are where the 2 curves make their sharpest turn.

The constant difference is the distance between the 2 vertices.

The asymptotes are not part of the Hyperbola but show where the curve would go in infinity.

Question 11

What is the equation for the asymptotes of a Hyperbola?

Answer 11

If it is centered at the origin, one vertex is a t (a,0) and the other is at (-a,0)

The 2 asymptotes are:

y = (^b/_a)x and

y = -(^b/_a)x

where b is the y-coordinate of any point on the branches and a is the x-coordinate.

Question 12

Therefore, what is the equation for the Hyperbola?

Answer 12

^x²/_a² - ^y²/_b² = 1

Question 13

How does this vary from the equation for an ellipse?

Answer 13

ellipse: ^x²/_a² + ^y²/_b² = 1
* ^{Think of an ellipse as an imploded hyperbola.}*

Question 14

How do you measure the eccentricity of a Hyperbola?

Answer 14

Eccentricity measures the ratio of the difference from the focus to the distance to the directrix

Question 15

What is the equation for eccentricity?

Answer 15

e = ^{√(a² + b²)}/_a

where a is the x-coordinate of any point and b is its y-coordinate.

Question 16

How does eccentricity work at the vertex?

Answer 16

1 /1 + 0/1 = 1

Question 17

What is the Latus Rectum of a Hyperbola?

Answer 17

The line which goes through the focus parallel to the Directrix.

Its length is ^2b²/_a

_{(= 2*y-coordinate/x-coordinate)}

Question 18

Is a Hyperbola a function?

Answer 18

No. It does not pass the Vertical Line test.

Question 19

Describe the parameters of the Hyperbola:

y²/25 - x²/36 = 1

Answer 19

The branches are top and bottom, not left and right.

a = 5; b= 6

e = ^{√(a² + b²)}/_a = 1.56

Question 20

What are the vertices and asymptotes of:

81x² - 9y² = 729

Answer 20

81x² - 9y² = 729

^81x²/₇₂₉ - ^9y²/₇₂₉ = 1

x²/9 - y²/81 = 1

a = 3, b = 9

asymptotes = ±9/3x

vertices = ±3,0

Question 21

Is this a parabola or a hyperbola:

x² + 5y = 7x

Answer 21

Parabola.

There is no y² term.

Question 22

Is this a parabola or a hyperbola:

3x² - 4y² = 60

Answer 22

Hyperbola.

^3x²/₆₀ - ^4y²/₆₀ = 1

Question 23

Is this a parabola or a hyperbola:

2y² = 4x² + 100

Answer 23

Hyperbola.

-4x² + 2y² = 100

^y²/₅₀ - ^x²/₂₅ = 1

Question 24

Does this hyperbola open along the x or y axis?

^x²/₁₆ - ^y²/₄₉ = 1

Answer 24

It opens along the x-axis.

Its vertices and (4,0) and (-4,0)

center = origin

Question 25

Does this hyperbola open along the x or y axis?

^y²/₉ - ^x²/₆₄ = 1

Answer 25

the y-axis

the vertices are (0,3) and (0,-3)

center = origin

Question 26

Does this hyperbola open along the x or y axis?

^y²/₈₁ - ^x²/₂₅ = 1

Answer 26

y-axis

vertices = (0,9) and (0,-9)

center = origin

Question 27

Describe the graph of ^x²/₁₂₁ - ^y²/₈₁ = 1

Answer 27

It opens along the x-axis.

Its vertices are (11,0) and (-11,0)

Its center is the origin

with a = 11 and b = 9 it is a fairly “square” hyperbola.

Question 28

Describe the graph of:

^x²/₁₆ - ^y²/₄₉ = 1

Answer 28

opens on the x-axis

vertices = (4,0) and (-4,0)

centers on origin

Question 29

Describe the graph of:

y² - x²/16 = 1

Answer 29

it opens along the y-axis

y²/1 - x²/16 = 1

vertices (0,1) and (0,-1)

This is a flat hyperbola.

Question 30

Describe the graph of:

y²/64 - x²/16 = 1

Answer 30

it opens along the y-axis

the vertices are (0,8) and (0,-8)

This is a tight parabola (?)