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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 / [email protected]
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 / [email protected]
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 / [email protected]
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 / [email protected]
PRECALCULUS PREPARATION LESSON Spring, 2015 TOPICS = CIRCLES & ELLIPSES Mr. Carlos Ortiz | ALAIN VILLASECA page 5 of 14
Copyright 2015 Carlos Ortiz / 305.785.5005 / [email protected]
PRECALCULUS PREPARATION LESSON Spring, 2015 TOPICS = CIRCLES & ELLIPSES Mr. Carlos Ortiz | ALAIN VILLASECA page 6 of 14
Copyright 2015 Carlos Ortiz / 305.785.5005 / [email protected]
PRECALCULUS PREPARATION LESSON Spring, 2015 TOPICS = CIRCLES & ELLIPSES Mr. Carlos Ortiz | ALAIN VILLASECA page 7 of 14
Copyright 2015 Carlos Ortiz / 305.785.5005 / [email protected]
PRECALCULUS PREPARATION LESSON Spring, 2015 TOPICS = CIRCLES & ELLIPSES Mr. Carlos Ortiz | ALAIN VILLASECA page 8 of 14
Copyright 2015 Carlos Ortiz / 305.785.5005 / [email protected]
PRECALCULUS PREPARATION LESSON Spring, 2015 TOPICS = CIRCLES & ELLIPSES Mr. Carlos Ortiz | ALAIN VILLASECA page 9 of 14
Copyright 2015 Carlos Ortiz / 305.785.5005 / [email protected]
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 / [email protected]
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 / [email protected]
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 / [email protected]
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 / [email protected]
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 / [email protected]
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