Introducing Conic Sections

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1. From each of following diagrams identify the conic formed and state how each is formed by
the intersection of a plane and the double-napped cone.
a. b. c.
2. The following are the locus definitions for different conics. List the correct conic for each
locus definition.
a. the absolute value of the difference of the distances from 2 given points (the 2 foci) is always
constant:
b. the sum of the distances from 2 given points (the 2 foci) is always constant:
c. the set of points at an equal distance from a given point:
d. the set of points at an equal distance from a point (the focus) and a line (the directrix):
3. A plane intersects a double-napped cone to form a circle. Assume the plane moves parallel to
its original position. Describe what happens to the circle that is formed when the plane
a) moves further away from the vertex
b) is at the vertex of the double-napped cone.
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c) moves closer to the vertex
Lesson Two
Graphing Circles and Rectangular Hyperbolas
1. Graph the following conics on the same grid.
2 2 Ax y . 1 + =
2 2 Bx y . 9 + =
2 2 Cx y . 36 + =
2. State the center and the radius for each of the following
circles.
a. 2 2 x y + = 4
b. 2 2 x y + =11
c.
2 2 16
25 x y + =
3. Write the equation of a circle with a center at (0,0) and a radius of 20 .
4. Write the equation of a circle with a center at (0,0) and a radius of 9
2
.
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5. Write the equation of a circle with a center at (0,0) and a diameter that is
8 units long.
6. Write the equation of a circle where the endpoints of the diameter are at
(6,7) and (-6, -7).
7. Write the equation of a hyperbola that has a center at (0,0) vertices at (1,0) and (-1,0) and the
equation of one asymptote is y x = -3 .
8. Prove that the point (2,3) is not on the graph of the hyperbola 2 2 4 9 36 x y – = –
9. Graph the hyperbola 2 2 x y – = -36 . Show the
asymptotes on your graph.
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Lesson Three
Expanding and Compressing Graphs of Conic Sections
1. Explain how an ellipse is different from a circle in terms of transformations.
2. Convert the following ellipse to standard form: 2 2 4 9 36 x y + =
3. State the length of the major axis and the minor axis, and the coordinates of the vertices for the
following ellipse. State if the major axis is vertical or horizontal: 2 2 25 16 400 x y + =
4. Sketch the graph for the following ellipse: 2 2 16 64 x y + =
5. Sketch the graph of each of the following parabolas. Label the coordinates of the vertex, and
the coordinates of two other points.
a.
1 2 3
2 y x = – –
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b. 2 y x = + 4 1
c. 2 x y = – – 2 3
6. Describe the direction of opening and the equation of the axis of symmetry for the following
parabolas.
a. y = 2×2
b. x = −7y2
c. y = − x2 −3
d. x = 4
5
y2 +2
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7. A parabola is symmetric about the x-axis and passes through the point (5, 2). What must be
another point on the parabola?
Lesson Four
Translating Graphs of Conic Sections
1. State the coordinates of the center for each of the following conics listed below.
a. (x +4)
2 + y2 = 36
b. 4(x −7)
2 +16(y +4)
2 = 64
2. State the coordinates of the vertices for each of the following conics.
a. 2 2 ( 1) 4( 3) 36 x y + – – = – b.
2 3 2 9( 1) 49 2 x y
æ ö ç ÷ + – + = è ø
3. Sketch the graph each for each of the following conics.
a. 2 2 ( 1) ( 3) 20 x y – ++ =
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b. 2 2 25( 2) 9( 4) 225 x y – – + = –
4. Write the equations for each of the following conics.
a. Parabola with vertex at (-3, 2), horizontal axis of symmetry, and passing through the point
(-5, 4).
b. Hyperbola that has a horizontal transverse axis 6 units long, and a conjugate axis 20 units long.
Hyperbola has been translated 1 unit up and 4 units right from (0,0).
Lesson Five
The Equation of a Conic Section in General Form
1. Change each of the following equations into general form.
a.
2 2 ( 1) ( 2) 1
4 3
x y – +
+ =
b.
2 2 ( 3) ( 3) 1
2 5
x y – + – = –
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2. Which conic could be represented by each equation. Explain your reasoning.
a. 2 2 4 6 3 12 24 0 xyxy + + – – =
b. 2 2 4 2 50 y xy – – + =
3. Which conic could be represented by the equation, 2 2 Ax Cy Dx Ey + ++= 0 given the following
conditions:
a. A C = ¹ 0
b. AC and A C >0 ¹
4. Determine the restrictions on the constants (parameters) A, C, D, and E in the equation
2 2 Ax Cy Dx Ey + ++= 0 given the following conditions on each conic listed:
a. A parabola with a vertical axis of symmetry with a vertex at (0, 0)
b. A circle with a center at a point not on the x or y axis.
5a. Given the conic
2 2
1
4 9
x y + = , write the new equation in standard form after a horizontal
expansion by a factor of 4 and a vertical compression by a factor of 1/3.
b. Given the conic
2 2
1
25 16
x y – = , write the new equation in standard form after a horizontal
compression by a factor of 1/2 and a vertical compression by a factor of 1/8.
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Lesson Six
The Equation of a Conic Section in Standard Form
1. Convert from general to standard form then answer the following questions:
a. 2 2 2 3 12 12 10 0 xy xy + — = Find the length of the major and minor axes
b. 2 2 4 4 12 30 0 xy x + – – = Find the center and length of diameter
c. 2 2 36 64 108 128 431 0 xy xy + + – – = Find the vertices
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ANSWERS
Lesson One Introducing Conic Sections
1. a) Circle – when a plane intersects a double-napped cone such that the plane is parallel to the
axis or perpendicular to the axis.
b) Parabola – when a plane intersects a double-napped cone such that the plane is parallel to the
generator.
c) Ellipse – when a plane intersects a double-napped cone such that the plane is neither
perpendicular nor parallel to the axis and the angle of intersection is greater than the generator
angle.
2a) Hyperbola b) Ellipse c) Circle d) Parabola 3. a) The radius of the circle increases so the
circle is larger b) The radius of the circle becomes infinitely small so a point is formed
c) The radius of the circle decreases so the circle is smaller
Lesson Two Graphing Circles and Rectangular Hyperbolas
2a. (0,0) 2 b. (0,0) 11 c. 4 (0,0) 5
1.
3. x2 + y2 = 20 4. x2 + y2 = 81
4
5. x2 + y2 =16
6. 2 2 x y + =85 7.
2
2 1
9
y x – =
8.
2 2 4(2) 9(3) 36
16 81 36 – = – – ¹-
9.
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Lesson Three Expanding and Compressing Graphs of Conic Sections
1. An ellipse is a circle that has either been horizontally and vertically compressed or expanded.
2.
2 2
1
9 4
x y + = 3. Major axis:10 Minor axis: 8 Vertices:(0,5) and (0,−5),Major axis vertical
4. 5. a)
b)
5. c)
6. a) Direction of opening: up Equations of axis of symmetry: x = 0
b) Direction of opening: left Equations of axis of symmetry: y = 0
c) Direction of opening: down Equations of axis of symmetry: x = 0
d) Direction of opening: right Equations of axis of symmetry: y = 0
7. (5, -2)
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Lesson Four Translating Graphs of Conic Sections
1a) (−4,0) b) (7,−4) 2a) ( 1,6) and ( 1,0) – – b) 11 17 , 1 and , 1 2 2
æ öæ ö ç ÷ç ÷ – — è øè ø
3. a) b)
4a) 1 2 3 ( 2) 2 x y + = – – b)
Lesson Five – The Equation of a Conic Section in General Form
1a) 2 2 3 4 6 16 7 0 xyxy + – + += b) 2 2 5 2 30 12 37 0 xy x y — – + =
2.a) Ellipse as A C and 0 ¹ AC > b) Parabola as A=0 and C ≠ 0 3a) Circle b)Ellipse
4a) A 0, C 0, D 0 E 0 ¹ = = ¹ b)A C, D 0, E 0 = ¹ ¹ 5a)
2
2 1
64
x
+ = y b)
2 4 2 4 1
25
x – y =
Lesson Six – The Equation of a Conic Section in Standard Form
1a)
b)
2 3 39 3 2 Center: ,0 Diameter: 39
24 2 x y
æ ö æö ç ÷ ç÷ – + = ® è ø èø
c)
2
2
3
2 ( 1) 11 5 1 Vertices: ,1 and ,1 16 9 2 2
x
y
æ ö ç ÷ + è ø – æ ö æö + = ® -ç ÷ ç÷ è ø èø

Sample Solution

The Positive Influence of Martial Arts

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Present day progress has, from numerous points of view, made people more vulnerable than proposed essentially. Regular daily existence has gotten progressively advantageous, and numerous perils are presently missing—individuals have less should be bold notwithstanding outside affliction. Present day people has become less portable and even leeway. The present human populace is by all accounts in a consistent move towards an absence of action that would keep them fit and ready to go to bat for themselves in the condition of a human threat.meaning

I recollect one case, from my previous school days, which affected my improvement later. Two 16-year-old domineering jerks attempted to take cash from a kid around 12 years of age. They conversed with him in an impolite way, however he remained quiet and didn’t react to their affront. At long last, one of the harassers made an endeavor to snatch the unfortunate casualty’s coat and hit him, however unexpectedly, the kid played out two or three momentary moves, and the two greater folks tumbled to the floor in tears. That was the first occasion when I saw hand to hand fighting in real life. Afterward, I warmed up to that kid, and it was a direct result of his eagerness about hand to hand fighting that I went into it myself. Presently I am certain this is one of the most valuable exercises for a cutting edge person.

There are an immense number of military fine arts. We utilize this term essentially when discussing eastern self-preservation and battling rehearses, which are umpteen, however, pretty much every nation on the planet has its own particular battle style (Porter 457). Random, one can name wushu (regularly erroneously called kung fu), capoeira, muay thai, kurash, jiu jitsu, karate, iaido, boxing, judo, Greco-Roman wrestling, ninjutsu, kick-boxing, and numerous others. Some of them are pretty much famous, however because of the procedures of globalization, one can concentrate practically any military artistic expression; so it’s anything but an amazement to see an African-American contemplating baguazhang, or a Japanese kid going into boxing. In any case, in spite of the wide assortment of battle rehearses, they all, to my brain, influence one’s character in a few significant manners.

The most evident advantage of rehearsing hand to hand fighting, regardless of whether it is wushu, karate, aikido, or whatever else, is physical advancement. A readied body is a significant condition to prevail right now. For example, taekwondo requires a person to have solid and very much extended legs to play out all the amazing procedures of this workmanship. Or then again, even a more splendid model, wushu xingyiquan, where the fingertips are utilized as a weapon. Envision the unbelievable shrouded conceivable outcomes of a human body, if even such an apparently frail piece of it as fingers can perform hazardous blows. Also, physical activities are very helpful in the event that one goes through the majority of the day in the workplace, sitting before a PC with a bowed back and stressed eyes (Porter 354). A few nights seven days spent in a dojo will have a huge and positive wellbeing impact.

The other, and maybe considerably progressively significant side going into combative techniques is a solidifying of one’s soul. Objections about troubles and agony are a somewhat basic occurring in preparing rooms. Numerous individuals quit preparing after their first hit in the face, or significantly prior, when they find a good pace combative techniques isn’t just about waving hands richly, yet additionally about difficult work and diligence. It is a lot simpler to give up and locate some reasonable reason than to go on, yet achievement can’t be accomplished in a flash. Numerous lesser triumphs signify a greater one, etc; and one day you notice you can do developments that appeared to be outlandish before all else. In this manner, hand to hand fighting foster tolerance and a feeling of direction, building up an ability to defeat challenges (Recli 65). These aptitudes would then be able to be applied as a rule, and I think this is one reason why eastern practices are regularly firmly identified with theory.

Maybe the most well-known motivation behind why individuals go into hand to hand fighting is a longing to increase fearlessness. They need to have the option to safeguard themselves, similarly as in the previously mentioned model with the 12-year-old kid, or which is watched much more frequently, to ensure their friends and family. Physical quality itself can be useful, however it can do nothing without soul (Recli 422). What’s more, regardless of whether one has both quality and soul, they could get more prominent advantages on the off chance that they ace certain aptitudes. Tragically, numerous individuals guided by their feeling of inadequacy, old complaints, or different injuries consider military to be as an approach to stand up for themselves by virtue of the powerless. These individuals frequently worry about a concern inside them, and as opposed to attempting to take care of their issues, they pick the way of a fanciful force, which gives them their very own feeling esteem. Thusly of reasoning turns into dead end, as it frequently brings about a solid and dexterous warrior with a similar frail soul as first and foremost, which isn’t valuable to anybody.

With respect to me, I encountered these impacts for myself. In spite of the fact that I am not a specialist, I am certain the previously mentioned contentions identify with practically any military work of art on the planet. To my psyche, one can’t ace an order in the event that one doesn’t summon certain adjustments in oneself; one needs to change, or quit. Be that as it may, the individuals who rehash themselves procure the products of their difficult work.

References

Doorman, William. Combative techniques Advantages. Los Angeles: Reality Sets Press, 2008. Print.

Recli, Sarah. The Spirit of Martial Arts Amidst Modernity. Brighton: Yin-Yang Books, 2011. Print.

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