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

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