# Additional Ellipse-circle Problems - March_12_2015 - Alain Villaseca

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eLLIPSE cIRCLE

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• PRECALCULUS PREPARATION LESSON Spring, 2015 TOPICS = CIRCLES & ELLIPSES Mr. Carlos Ortiz | ALAIN VILLASECA page 1 of 14

Copyright 2015 Carlos Ortiz / 305.785.5005 / carlos@mrcarlosortiz.com

1 Finding the STANDARD FORM OF THE EQUATION OF AN ELLIPSE from given

information:

Find the standard form of the equation of the ellipse having foci (0,1) and (4,1) and a major axis of length 6, as shown

in Figure. (Hint: Alain, Ive indicated a and b in the figure and plotted the center of the ellipse).you find them and youll be at your answer in no time)

PRECALCULUS PREPARATION LESSON CONICS: ELLIPSE & CIRCLE

Course: PRECAL Preparation Delivered: Thursday, March 12, 2015

Instructor: Mr. Carlos Ortiz To Be Completed By: Thursday, March 12, 2015

Student: ALAIN VILLASECA ETC: 180 min

The trouble with quotes on the internet is that you never know if they are genuine.

Abraham Lincoln

• PRECALCULUS PREPARATION LESSON Spring, 2015 TOPICS = CIRCLES & ELLIPSES Mr. Carlos Ortiz | ALAIN VILLASECA page 2 of 14

Copyright 2015 Carlos Ortiz / 305.785.5005 / carlos@mrcarlosortiz.com

2 COMPLETING THE SQUARE to find the STANDARD FORM OF THE EQUATION OF AN ELLIPSE and then SKETCHING IT:

Find the standard form of the equation of the ellipse having foci (2,0) and (2,6) and a

major axis of length 8. Then, SKETCH the ellipse.

3 WRITING THE EQUATION OF A CIRCLE:

The point (3,4) lies on a circle whose center is at (-1,2), as shown here:

Write the standard form of the equation of this circle:

• PRECALCULUS PREPARATION LESSON Spring, 2015 TOPICS = CIRCLES & ELLIPSES Mr. Carlos Ortiz | ALAIN VILLASECA page 3 of 14

Copyright 2015 Carlos Ortiz / 305.785.5005 / carlos@mrcarlosortiz.com

4 WRITING THE EQUATION OF A CIRCLE:

The point (1,-2) lies on a circle whose center is at (-3,-5). Write the standard form of

the equation of this circle.

Write the standard form of the equation of this circle:

• PRECALCULUS PREPARATION LESSON Spring, 2015 TOPICS = CIRCLES & ELLIPSES Mr. Carlos Ortiz | ALAIN VILLASECA page 4 of 14

Copyright 2015 Carlos Ortiz / 305.785.5005 / carlos@mrcarlosortiz.com

• PRECALCULUS PREPARATION LESSON Spring, 2015 TOPICS = CIRCLES & ELLIPSES Mr. Carlos Ortiz | ALAIN VILLASECA page 5 of 14

Copyright 2015 Carlos Ortiz / 305.785.5005 / carlos@mrcarlosortiz.com

• PRECALCULUS PREPARATION LESSON Spring, 2015 TOPICS = CIRCLES & ELLIPSES Mr. Carlos Ortiz | ALAIN VILLASECA page 6 of 14

Copyright 2015 Carlos Ortiz / 305.785.5005 / carlos@mrcarlosortiz.com

• PRECALCULUS PREPARATION LESSON Spring, 2015 TOPICS = CIRCLES & ELLIPSES Mr. Carlos Ortiz | ALAIN VILLASECA page 7 of 14

Copyright 2015 Carlos Ortiz / 305.785.5005 / carlos@mrcarlosortiz.com

• PRECALCULUS PREPARATION LESSON Spring, 2015 TOPICS = CIRCLES & ELLIPSES Mr. Carlos Ortiz | ALAIN VILLASECA page 8 of 14

Copyright 2015 Carlos Ortiz / 305.785.5005 / carlos@mrcarlosortiz.com

• PRECALCULUS PREPARATION LESSON Spring, 2015 TOPICS = CIRCLES & ELLIPSES Mr. Carlos Ortiz | ALAIN VILLASECA page 9 of 14

Copyright 2015 Carlos Ortiz / 305.785.5005 / carlos@mrcarlosortiz.com

SOLUTIONS:

1 Finding the STANDARD FORM OF THE EQUATION OF AN ELLIPSE from given information:

Find the standard form of the equation of the ellipse having foci (0,1) and (4,1) and a

major axis of length 6, as shown in Figure.

• PRECALCULUS PREPARATION LESSON Spring, 2015 TOPICS = CIRCLES & ELLIPSES Mr. Carlos Ortiz | ALAIN VILLASECA page 10 of 14

Copyright 2015 Carlos Ortiz / 305.785.5005 / carlos@mrcarlosortiz.com

2 COMPLETING THE SQUARE to find the STANDARD FORM OF THE EQUATION OF

AN ELLIPSE and then SKETCHING IT:

Find the standard form of the equation of the ellipse having foci (2,0) and (2,6) and a major axis of length 8. Then, SKETCH the ellipse.

• PRECALCULUS PREPARATION LESSON Spring, 2015 TOPICS = CIRCLES & ELLIPSES Mr. Carlos Ortiz | ALAIN VILLASECA page 11 of 14

Copyright 2015 Carlos Ortiz / 305.785.5005 / carlos@mrcarlosortiz.com

3.

4.

2

2

2

22

2 2

2

2

2 2

2

2

2

2

2 2

Find r from the "distance formula" and the points (1,-2) and (-3,-5):

1 3 2 5

1 3 2 5

4 3

16 9

so since CENTER ( 3, 5), we have

25

3 5 25

r

r

r

r

x h

r

x y

y k r

70.

75.

76.

2

2

2 2

2

Center is midpoint of diameter:

-4 + 4 -1 + 1C , =

2 2

radius half of diameter which we find

with our distance formula:

4 4 1 1

64 4

68

4 17 2 17

2 1717

2

(0,0)

17

, :

2

d

d

d

d

dso radius r

and fina r

SO final equation is

lly

x y

17

• PRECALCULUS PREPARATION LESSON Spring, 2015 TOPICS = CIRCLES & ELLIPSES Mr. Carlos Ortiz | ALAIN VILLASECA page 12 of 14

Copyright 2015 Carlos Ortiz / 305.785.5005 / carlos@mrcarlosortiz.com

81.

• PRECALCULUS PREPARATION LESSON Spring, 2015 TOPICS = CIRCLES & ELLIPSES Mr. Carlos Ortiz | ALAIN VILLASECA page 13 of 14

Copyright 2015 Carlos Ortiz / 305.785.5005 / carlos@mrcarlosortiz.com

5. B

6. C 7. A

8. D

9.

2 2

2 2

2 2

Center: (0,0)

4, 2

Vertical major

14 16

axis

1

a b

x h y k

b a

x y

10.

2 2

2

2 2

2

Center: (0,0)

32,

2

Horizontal major

194

4

axis

1

a b

x h y k

b

x y

a

11.

2 2

Alain, this one is "ALMOST"a circle...if

2 2 2

2

2

2 2

2 2

2

the "45" had beena "49" instead, we would ha

(0,0

2

e

r

v :

)x + y = 49

Cente

149 4

r: (0,0)

7, 2

4 49

45

Horizontal major axis

5

1

this is a circle centered atwith a

a c

c a b

b

b

x y

x h y k

a b

adius = 7

• PRECALCULUS PREPARATION LESSON Spring, 2015 TOPICS = CIRCLES & ELLIPSES Mr. Carlos Ortiz | ALAIN VILLASECA page 14 of 14

Copyright 2015 Carlos Ortiz / 305.785.5005 / carlos@mrcarlosortiz.com

18.

2

2

2 2

2 2

2 2

2 2

2

2 2

2

2

2

2

Center: (0,0) and a 5 so

x - h y - k+ = 1

b a

x y+ = 1

b

SINCE it goes thru ( ), then

+ = 1b

16 4+ = 1

b 25

16 21

b 25

21b

a 25

25

25

400

4,2

4 2

b21

x

400

so,

, + = 1400 25

21

thereforey