Conic Sections

Question 1

Name each
conic section.

Answer 1

Circle

Ellipse

Parabola

Hyperbola

Question 2

What
information
do you need to write the
equation of a
circle in standard form?

Answer 2

• Center (h, k)
• and either
  
  • Radius
• or*
  
  • Another Point on the Circle

Question 3

What is the

• *standard equation** of a
• *circle**?

Answer 3

(x − h)² + (y − k)² = r²

h: x-coordinate of center
k: y-coordinate of center
r: radius length

Question 4

Given a
circle’s

center at
(−3, 4)

and
radius length of
7,

what is the circle’s
equation in standard form?

Answer 4

(x + 3)² + (y − 4)² = 49

Question 5

Given a
circle’s

center at
(1, 1)

and that

(3, 4)
lies on the circle,

what is the circle’s
radius?

Answer 5

√13

Apply the distance formula

r = √( Δx² + Δy²)

= √ ((3 − 1)² + (4 − 1)²​)
(it doesn’t matter which point goes first in the subtraction as long as you’re consistent)

= √ ((2)² + (3)²​)

= √ ((2)² + (3)²​)

= √ (4 + 9​)

= √13

Question 6

What is the

• *expanded form** of a
• *circle**?

Answer 6

The result of
expanding the binomial squares
in the standard form and
combining like terms

e.g.:
x² + y² − 2x − 4y − 4 = 0

Question 7

Given this
circle’s equation in
expanded form,​

x² + y² − 2x − 4y − 4 = 0,

how do you

• *write** the equation in
• *standard form**?

Answer 7

Complete the squares

x² + y² − 2x − 4y − 4 = 0

x² − 2x + y² − 4y − 4 = 0

x² − 2x + y² − 4y = 4

(x² − 2x + 1) + (y² − 4y + 4) = 4 + 4 + 1
(you know which numbers to add by halving each of the first-degree terms and then squaring that result)

(x − 1)² + (y − 2)² = 9

Question 8

What is the

• *standard equation** of an
• *ellipse**?

Answer 8

(x − h)² + (y − k)² = 1
_{<span>a</span>}² b²

h: x-coordinate of center
k: y-coordinate of center
a: horizontal radius (think of as r_h)
b: vertical radius (think of as r_v)

Question 9

What are the
noteworthy points
of an
ellipse?

Answer 9

Center

Foci

Vertices

Co-vertices

Question 10

What point is
shown here?

Answer 10

The

• *center** of an
• *ellipse**

Question 11

What points are
shown here?

Answer 11

The

• *foci** of an
• *ellipse**

Question 12

What points are
shown here?

Answer 12

The

• *vertices** of an
• *ellipse**

Question 13

What points are
shown here?

Answer 13

The

• *co-vertices** of an
• *ellipse**

Question 14

What are the
noteworthy distances
of an
ellipse?

Answer 14

Major radius

Minor radius
(q)

Focal length
(f )

(p)

Question 15

In the
ellipse below,
what
distance is shown?

Answer 15

The
major radius

The distance from the
center to
either vertex

Usually called p

Question 16

In the
ellipse below,
what
distance is shown?

Answer 16

The
minor radius

The distance from the
center to
either co-vertex

Usually called q

Question 17

In the
ellipse below,
what
distance is shown?

Answer 17

The
focal length

The distance from the
center to
either focus

Usually called f

Question 18

From any

• *point** (N) on an
• *ellipse**, the combined
• *distances** to the
• *foci** (f₁ and f₂) is equal to what?

Answer 18

d(N, f₁) + d(N, f₂) = 2p

• Or two times the length of the major radius*
• How we know this:*
• ​Vertices (v₁ and v₂, in orange) each lie along the major axis at a distance of p away from the center, so they are symmetrical about the center
• Foci (f₁ and f₁, in red) each lie along the major axis at a distance of f away from the center, so they are symmetrical about the center
• Therefore, the red lengths are congruent and the orange lengths are congruent
  f = f
  p = p
• The vertices lie on the ellipse, so by definition, the distance from v₁ to f₁ plus the distance from v₁ to f₂ is equal to some constant amount (k):
  d (v₁, f₁) + d (v₁, f2) = k
• d (v₁, f₁) = p − f
  d (v₁, f2) = 2f + p − f
• k is equal to 2f (the red length) plus 2(p − f ) (the orange amount)
  k = 2f + p − f + p − f
  = 2p

Question 19

In an
ellipse,
what’s the
relationship between the
major radius, minor radius, and
focal length?

Answer 19

f ² = p² − q²

Derived from the
Pythagorean theorem

With any two pieces of information,
you can calculate the third

How we know this:

• d (c₁, v₁) + d (c₁, v₂) = 2p
• Co-vertices (c₁ and c₂) are equidistant from the foci
• d* (c₁, v₁) = d (c₁, v₂)
• 2d (c₁, v₁) = 2p
• d* (c₁, v₁) = p
• q is purple; f is red; p is orange
• q² + f ² = p²
• f *² = p² − q²

Question 20

What is the

• *center** of this
• *ellipse**?

Answer 20

(4, −6)

The standard equation of an ellipse is

(x − h)² + (y − k)² = 1
r_h² r_v²

where (h, k) is the center of the ellipse, so the coordinates both switch signs

Question 21

What is the

• *major radius** of this
• *ellipse**?

Answer 21

p = 3

• The major radius, or p, is the longer of the ellipse’s two radii*
• When an ellipse is in standard form, the radius lengths are squared​*

Question 22

What is the

• *minor radius** of this
• *ellipse**?

Answer 22

q = 2

• The major radius, or p, is the longer of the ellipse’s two radii*
• When an ellipse is in standard form, the radius lengths are squared*

Question 23

An
ellipse’s
center is (−1, 1),
major radius is 6,
minor radius is 4, and
vertices are (−1, 7) and (−1, −5).

How would the ellipse be
written in standard form?

Answer 23

Question 24

An
ellipse’s
center is (0, 0),
major radius is 5,
minor radius is 4, and
co-vertices are (4, 0) and (−4, 0).

How would the ellipse be
written in standard form?

Answer 24

Question 25

An
ellipse’s
center is (0, 0),
major radius is 5,
minor radius is 4, and
co-vertices are (4, 0) and (−4, 0).

Where are the ellipse’s
vertices?

Answer 25

(0, 3) and (0, −3)

f² = p² − q²

= 5² − 4²

= 25 − 16

= 9

f = √9

f = 3

• Foci lie on the major axis, which includes the center at (0, 0) and the vertices at (0, 5) and (0, −5)*
• Foci are at a distance of 3 away from the center*
• Foci are at (0, 3) and (0, −3)*

Question 26

Graphically,
parabolas can be viewed
as the set of
all points whose
distance from
_____ is
equal to their distance from

_____.

Answer 26

a certain point

and a

certain line
(the directrix)

(the focus)

Question 27

Given a
parabola’s
focus at (6, −4) and
directrix at y = −7,
how would you write its
equation?

Answer 27

1. Write an equation
   distance to focus = distance to directrix
   for any point (x, y) on the parabola
2. Solve for the first-degree term

√((y + 7)²) = √( (x − 6)² + (y + 4)²)

(y + 7)²​ = (x − 6)² + (y + 4)²

y² + 14y + 49 = (x − 6)² + y² + 8y + 16

14y − 8y = (x − 6)² + 16 − 49

6y = (x − 6)² − 33

y = (x − 6)² − 11
6 2

Question 28

What is the

• *equation** for the
• *parabola** below in terms of its
• *focus and directrix**?

y = _____

Answer 28

1 • (x − a)² + 1 • (b + k)
2 (b − k) 2

a: x-coordinate of focus
b: y-coordinate of focus
k: directrix

Question 29

What is the

• *standard equation** of a
• *hyperbola**?

Answer 29

(x − h)² − (y − k)² = 1
a² b²

or

(y − k)² − (x − h)² = 1
b² a²

h: x-coordinate of center
k: y-coordinate of center
a: horizontal radius (think of as r_h)
b: vertical radius (think of as r_v)

Question 30

What are the

• *major features** of
• *hyperbolas**?

Answer 30

1. Center
2. Vertices
3. Foci
4. Asymptotes
5. Directions
   (open up/down or left/right)

Question 31

What point is
shown here?

Answer 31

The

• *center** of a
• *hyperbola**

Question 32

What points are
shown here?

Answer 32

The

• *vertices** of a
• *hyperbola**

Question 33

What points are
shown here?

Answer 33

The

• *foci** of a
• *hyperbola**

Question 34

What is
shown here?

Answer 34

The

• *asymptotes** of a
• *hyperbola**

Question 35

Given hyperbola

(x + 1)² − (y − 2)² = 1
9 4

what will its
directions be?

Answer 35

Left/right

Because the x term is positive

Question 36

Given hyperbola

(y + 1)² − (x − 2)² = 1
9 4

what will its
directions be?

Answer 36

Up/down

Because the y term is positive

Question 37

Given hyperbola

x² − y² = 1
9 16

how do you determine the
focal length?

Answer 37

f ² = a² + b²

• Related to the ellipse*
• f *² = a² + b²

= 9 + 16

= 25

f = √25

f = 5

Question 38

From any

• *point** (N) on a
• *hyperbola**, the combined
• *distances** to the
• *foci** (f₁ and f₂) is equal to what?

Answer 38

| d(N, f₁) − d(N, f₂) | = 2p

Or two times the length from the
center to a
vertex

Question 39

How do you determine the

• *asymptotes** of a
• *hyperbola**?

Answer 39

Determine

• *what happens** as the
• *negative variable**
• *approaches ±∞**

Example:

y²/4 − x²/9 = 1

y²/4 = x²/9 + 1

y² = 4/9 • x² + 4

y = √(4/9 • x² + 4)

x → ±∞, y ≈ ±√(4/9 • x²)

y ≈ ±√(4/9 • x²)

y ≈ ±2/3 • x (these are the asymptotes)

Question 40

How would you

• *recognize** a
• *nonstandard equation** as a
• *circle**?

Answer 40

Both variables are in
second-degree terms
(unlike parabolas)

Coefficients of both variables is the
same
(unlike most ellipses)

Signs of the terms containing the variables is the
same
(unlike hyperbolas)

Question 41

How would you

• *recognize** a
• *nonstandard equation** as an
• *ellipse**?

Answer 41

Coefficients of the variables is probably
different
(unlike circles)

Both variables are in
second-degree terms
(unlike parabolas)

Signs of the terms containing the variables is the
same
(unlike hyperbolas)

Question 42

How would you

• *recognize** a
• *nonstandard equation** as a
• *parabola**?

Answer 42

• *One variable** in a
• *second-degree term**
• *One variable** in a
• *first-degree term**

(unlike circles, ellipses, and hyperbolas)

Question 43

How would you

• *recognize** a
• *nonstandard equation** as a
• *hyperbola**?

Answer 43

Both variables are in
second-degree terms
(unlike parabolas)

Signs of the terms containing both variables is
different
(unlike circles and ellipses)