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** ## Footnote 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 = *a*² + *b*²** * Related to the ellipse* * f *² = *a*² + *b*² = 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) | =** **2*p*** ## Footnote *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 ±∞** ## Footnote *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)