18 Circles Ellipses

Question 1

What are the set of all points a given distance from 1 point?

Answer 1

A circle

Question 2

What is the standard equation for a circle?

Answer 2

x² + y² = radius²

Question 3

What are ellipses?

Answer 3

the set of points on a plane such that the sum of the distances from 2 points (foci) is constant.

Question 4

What is the formula for an ellipse?

Answer 4

^x²/_a² + ^y²/_b² = 1 OR

^y²/_a² + ^x²/_b² = 1

where a is half the major axis and y is half the minor axis.

If a² is under the x², then it is a horizontal ellipse.

Question 5

How are a and b related?

Answer 5

a is always larger than b.

Question 6

Are circles and ellipses functions?

Answer 6

No. They don’t pass the vertical line test.

Question 7

What are the vertices of an ellipse?

Answer 7

the 2 endpoints of the major axis

Question 8

Describe this ellipse:

^x²/₉ + ^y²/₄ = 1

Answer 8

A horizontal ellipse centered on the origin with vertices at ±3,0

Question 9

Describe the ellipse:

^x²/₁₆ + ^y²/₂₅ = 1

Answer 9

A vertical ellipse with vertices at

(0, ±5)

Question 10

How do you find the foci of an ellipse?

Answer 10

a² - b² = c²

c² = the square of the distance from the center of each focus.

Question 11

What are the foci of:

^x²/₁₆ + ^y²/₂₅ = 1

Answer 11

25 - 16 = c²

The foci are at (±3,0)

Question 12

What is the eccentricity of:

x²/49 + y²/81 = 1

Answer 12

The eccentricity is the ratio of the length of the minor axis divided by the length of the major axis. The closer it gets to 1, the more it is like a circle:

14/18 = 7/9

Question 13

How do you get a 2D circle from a 3D cone?

Answer 13

a flat cross-section

Question 14

What is a circle a special case of?

Answer 14

an ellipse

Question 15

What is the radius of:

^x²/₃₆ + ^y²/₃₆ = 1 ?

Answer 15

^x²/₃₆ + ^y²/₃₆ = ³⁶/₃₆

x² + y² = 36

radius = 6

Question 16

What is the radius of

4x² = 36 - 4y² ?

Answer 16

4x² = 36 - 4y²

x² + y² = 9

radius = 3

Question 17

Is this a circle or an ellipse?

x²/100 + y²/121 = 1

Answer 17

ellipse

a and b are different

(it is a vertical ellipse - a is under y²)

Question 18

Is this a circle or an ellipse?

x²/100 + y²/100 = 1

Answer 18

a circle with radius 10

Question 19

Is this a circle or an ellipse?

6x² + 7y² = 42

Answer 19

6x² + 7y² = 42

x²/7 + y²/6 = 1

It is an ellipse.

Question 20

What is the eccentricity?

x²/9 + y²/169 = 1

Answer 20

minor axis = √9 * 2 = 6

major axis = √169 * 2 = 26

eccentricity = 3/13

Question 21

What is the eccentricity?

x²/81 + y²/64 = 1

Answer 21

minor axis = √64 * 2 = 16

major axis = √81 * 2 = 18

eccentricity = 8/9

Question 22

What is the eccentricity?

x²/1 + y²/64 = 1

Answer 22

minor axis = √1 * 2 = 2

major axis = √64 * 2 = 16

eccentricity = 1/8

Question 23

Describe:

x²/9 + y²/225 = 1

Answer 23

It is a vertical ellipse because a² is under y².

The center is the origin.

The vertices are (0, ±15)

The foci are √(225 - 9) =

(0, ±14.697)

Eccentricity is 1/5, very flat.

Question 24

Describe:

x²/225 + y²/9 = 1

Answer 24

It is a horizontal ellipse because a² is under x².

The center is the origin.

The vertices are (±15,0)

The foci are √(225 - 9) =

(±14.697, 0)

Eccentricity is 1/5, very flat.

Question 25

Describe:

x²/25 + y²/25 = 1

Answer 25

It is a circle, centered at the origin with a radius of 5.

Question 26

Describe:

25x² + 16y² = 400

Answer 26

x²/16 + y²/25 = 1

It is a vertical ellipse

vertices = (0, ±5)

foci = √(25 + 16) = √41

eccentricity = 8/10 = 4/5