Pre Calc Session 5 Stand Up Conic

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Pre Calc Session 5Stand Up Conic

• Review Activity

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Probably Parabola
Circle Cents
Statistical Elliptical
Hype the Hyperbola
Apply Here
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3

10 points

What is the focal distance for the parabola whose
equation in standard form is (y-5)2 -8(x-3) ?


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4
A N S W E R
10 points
4p -8 so p-2 meaning the focal distance is 2.
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5

20 points
If the end points of the latus rectum for a
parabola are (-1,11) and (-1,-1), then find its
vertex in quadrant I.
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6
A N S W E R
20 points
Since the vertex must be in quadrant I, its
coordinates will be (2,5). (It opens to the left
with p-3.)
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7

30 points
What is the equation of the directrix for the
parabola whose equation is (y3)2 2(x-4)
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8
A N S W E R
30 points
x 3.5 because it is a vertical line exactly a
half unit to the left of the vertex (4,-3) so
that it opens to the right.
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9

40 points
What are the coordinates for the focal point of a
parabola whose equation is y2 6y 4x 25 0
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10
A N S W E R
40 points
(5,-3), exactly 1 unit to the right of the vertex
at (4,-3). You need to complete the square to
transform the general form equation into standard
form (y3)2 4(x-4)
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11

50 points
Write the equation of a parabola whose directrix
is y 3.5 and one end point of the latus rectum
is (-2, .5).
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12
A N S W E R
50 points
(x-1)2 6(y-2) because the directrix is 1.5
units above the vertex y value which is 2 while
the latus rectum whose length is 6 will have its
other end point at (4, .5).
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13

10 points
What is the radius for a circle whose equation is
(x-5)2 (y8)2 10 ?
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14
A N S W E R
10 points
v10

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15

20 points
Two concentric circles are given by the equations
(x-3)2 (y-1)2 4 and (x-3)2 (y-1)2 64 If
the smaller of the two circles is a bulls-eye of
a dart board, then what is the probability of a
randomly thrown dart hitting the board being in
the bulls-eye portion of it?
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16
A N S W E R
20 points
Bulls-eye 4p Whole board 64p Probability ?

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17

30 points
Write the general form equation of a circle whose
center is at (4,-2) and whose radius is 7.
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18
A N S W E R
30 points
Standard (x-4)2 (y2)2 49 Expanded
x2-8x16y24y4 49 General x2 y2 - 8x 4y
29 0
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19

40 points
Write the equation of a circle for which a
diameter has end points P(-3,4) Q(2,1)
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20
A N S W E R
40 points
(x ½)2 (y 5/2)2 8.5

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21

50 points
Consider the conic whose general form equation is
x2 y2 - 2x 4y 5 0. Describe the graph of
this conic section.
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22
A N S W E R
50 points
In standard form it is (x-1)2 (y2)2 0 So it
seems to be a circle, but because the perceived
radius is 0, what the graph really becomes is a
single point. This is what is commonly called a
degenerate conic section.

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23

10 points
Describe how you determine if the major axis of
an ellipse runs parallel to the x axis
(horizontally) or y axis (vertically).
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24
A N S W E R
10 points
If the a2 (largest denominator in standard form)
term is under the x term, it is a wide ellipse
with the major axis horizontal, while if it is
under the y term, it is a skinny ellipse with a
vertical major axis.

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25

20 points
The major axis of an ellipse is 34 units while
its focal length is 8 units. What is the length
of its minor axis?
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26
A N S W E R
20 points
Since a2 b2 c2 and a 17 while c8, then b
must equal 15 units. This means the minor axis
is 30 units long.

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27
30 points

What is true about the eccentricity of all
ellipses. Explain why.
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28
A N S W E R
30 points
Since c lt a for an ellipse, then the ratio of c/a
will always be less than 1.

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29

40 points
The foci of an ellipse are (-2,0) and (2,0) while
the semi-major axis is 7 units long. Find the
equation of this ellipse.
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30
A N S W E R
40 points
x2
y2

1

45
49
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31

50 points
The end points of the 4 vertices of an ellipse
are (-8,3), (-1,6), (6,3), and (-1,0). What are
the coordinates for the focus found in quadrant
II?
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32
A N S W E R
50 points
Since c 2v10 , the focus in quadrant II will be
this distance to the left of the center point
(-1,3). This makes the focus at (-1-2v10, 3 )

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33

10 points
What is the center of a hyperbola whose conjugate
axis has endpoints at (-4,-2) and (-4, 4) ?
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34
A N S W E R
10 points
(-4,1) because the mid point of the conjugate
axis is the hyperbolas center. (So is the
midpoint of the transverse axis!)

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35

20 points
The eccentricity of a hyperbola is 1.5 for an
origin centered conic with a focus at (9,0).
Name a vertex of this hyperbola.
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36
A N S W E R
20 points
If c9 and e1.5, then a6 meaning the two
vertices are at (6,0) and (-6,0).

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37

30 points
A hyperbola is called equilateral if ab. (This
makes the slope of the asymptotes are 1 and -1.)
Write the equation of an equilateral hyperbola
with foci at (8,0) and (-8,0).
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38
A N S W E R
30 points
x2
y2
-

1
32
32
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39

40 points
Fill in the blanks. Definition of a
Hyperbola The locus of all points such that the
_________ in the distances from _____ given
_______ will remain constant.
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40
A N S W E R
40 points
difference 2 points
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41

50 points
Find the equations of the asymptotes for the
hyperbola whose equation is y2 5x2 20x 50
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42
A N S W E R
50 points
y v5 (x 2) a v30 and b v6 after
completing the square and putting this equation
into standard form.
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43

10 points
A parabolic dish at the Buffalo Bills game is
considered a deep dish because the ratio of the
focal length to its diameter is .2 Its focal
length is 8 inches. How much longer is the
diameter as compared to the latus rectum of this
parabola?
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44
A N S W E R
10 points
Since .2 is 8/40 , the diameter is 40 inches.
Since the latus rectum is 4x8 32 inches, the
diameter is 8 inches longer than the latus
rectum. (40- 32 8)

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45

20 points
Three circular rollers are tangent to each other
and their centers are all in a line. The center
roller has a radius of 25 while the others each
have a radius of 5. How many inches are between
the centers of the two smaller rollers? How many
rotations does each make for the middle roller to
rotate once? Will both small rollers rotate the
same direction?
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46
A N S W E R
20 points
As roller 1 goes clockwise, roller 2 will go
counterclockwise rotating 1 time for every 5
times each of the smaller rollers rotate. This
will cause the 3rd roller to go clockwise just as
the first roller had rotated.

60
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47

30 points
The bulb of a flashlight is located at the focus
of a parabolic reflecting surface. The vertex of
the parabola to the lens surface is 5 cm, with
the bulb exactly 40 of the way to the lens. What
is the diameter of this circular lens?
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48
A N S W E R
30 points
The equation might look like y2 8(x-0) So if x
5, the diameter is going to be 2v40 cm
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49

40 points
All of the planets in our solar system have
elliptical orbits with the sum as one focal
point. If Mercurys orbit is 46 million km from
the sun at its closest point and 70 million km
from the sun at its farthest point, what is the
length of the major axis?
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50
A N S W E R
40 points

116 million km
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51

50 points
Two Coast Guard stations at (0,0) and (0,1600),
1600 miles apart, North South of each other,
receive an SOS message to determine the sender is
200 miles closer to the station at the origin.
Write the equation of the hyperbola on which the
stranded ship must be located. Find the exact
distance to the origin station if you know the
ship is 600 miles East of it.
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52
A N S W E R
50 points
(y-800)2
x2
1
-


630,000
10,000
Ship is at (600,674.64) a distance of 902.9 miles
from the origin.
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