Chapter 1 Conic Sections

8/12/2019 Chapter 1 Conic Sections

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Name Matrix No Group

CHAPTERonic Sections

1 1Introduction to Conic Sections

Definition of conic sectionsConic sections are the curves which result from the intersection of aplane with a cone.

a) Circle

When a plane intersects a double-napped cone and is parallel to thebase of a cone, a circle is formed.


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b)Parabola

When a plane intersects a double-napped cone and is parallel to theside of the cone, a parabola is formed.

c) Ellipse

When a plane intersects a double-napped cone and is neither parallelnor perpendicular to the base of the cone, an ellipse can be formed.

d)Hyperbola

When a plane intersects a double-napped cone and is neither parallelnor perpendicular to the base of the cone, a hyperbola can be formed


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1 2CirclesDefinition of CirclesA circle is a set of all points in a plane equidistant from a given fixed

point called the center. A line segment determined by the center and anypoint on the circle is called a radius.

(

,

) = center of circle

= radius

Let the center of the circle b(, )e and the fixed distance be and for any point on the circle with coordinate (, )Using distance formula

= 2 + 22 = 2 + 2 [ Square both sides ]

Equation of circlea)Standard Form

2

= 2

+ 2

Center = , Radius = b)General Form

2 + 2 + 2+ 2+ = 0Center = , Radius = = 2 + 2

(, )

(, )


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Example 1:1. Find the equation of the circle having its center at 0,0and radius

of length 3 units.

2. Find the equation of a circle having its center at 3,5and radiusof length 4units.

3. Find the equation of a circle that has its center at 5, 9and aradius of length 23units.


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The centre and radius of a circle by completing squareTo purpose of completing the square is to convert the general equationof the circle into the standard form.

4. Graph 2 + 2 6+ 4+ 9 = 0

5. Find the center and the length of the radius of the circle2 + 2 6+ 12 2 = 0.


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6. Find the center and the radius of the circle with equation2 + 2 4 2= 4.

Circle passing through three given points7. Find the equation of the circle passing through the points (0,1).

(4,3), and (1,

1).


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10.Find the equation of the circle having as diameter where isthe point (1,8)andis the point (3,14).

The points of intersection of two circles11.Find the intersections between the circle 2 + 2 = 10 and line+ 2= 5.


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12.Find the coordinates of the points of intersection of the two circleswith equation

2 + 2 3+ 13 48 = 02 + 2 + 3= 0


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The equations of tangents and normal to a CircleTheorem 1Suppose we have a standard equation 2 + 2 = 2, so the equationof a tangent for the circle at the point of 1, 1is given by1+ 1= 2.

Figure 1

13.Find the equation of the tangent to a circle2 + 2 = 13 at the point 2,3.

1, 1

1+ 1= 22 + 2 = 2


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14.Find the equation of the tangent to a circle2 + 2 = 25 at the point 3,4.

Theorem 2If we have a common equation 2 + 2 + 2+ 2+ = 0, thetangent to the circle at the point 1, 1is

1+ 1+ + 1 + + 1 + = 015.Find tangent and normal line of the circle

2 + 2 6 10 82 = 0at the point 1, 5.


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16.Find tangent and normal line of the circle2 + 2 4 6+ 8 = 0at the point 1,5.

The length of the tangent to a CircleTheorem 3The length of the tangent from

fixed point

,

to a circle with

equation 2 + 2 + 2+ 2+ = 0(denote by ), is given by

= 2 + 2 + 2 + 2+ .

Figure 2

,

,


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17.Find the length of the tangent from the point 4,6to the circle2 + 2 4 2= 6

18.Find the length of the tangents from the point 8,4to the circlewith centre 3,0and radius 2.


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1 3ParabolaDefinition of ParabolaA parabola is defined as a set of points in a plane that are equidistant

from a fixed point (focus) and from a fixed line (directrix).(see figure 3)

Figure 3 Figure 4We obtain a particularly simple equation for a parabola if we place its

vertex as origin 0and its directrix parallel to the -axis as in Figure 4.If the focus is the point 0, then the directrix has the equation= . If , is any point on the parabola, then the distancefrom to the focus is

= 2 + 2and the distance from to the directrix is + .

The defining property of a parabola is that these distances are equal:

2 + 2 = + We get equivalent equation by squaring and simplifying:

2 + 2 = + 2 = + 2

2

+ 2

2+ 2

= 2

+ 2+ 2

2 = 4


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The equation of a Parabola with Vertex 0,0The standard equation for parabolas with vertex 0,0 is summarizedbelow.

Parabola Vertex Focus Directrix Shape

2 = 40,0 , 0 =

Opens to the right

0,0 , 0 = Opens to the left

2 = 40,0 0, =

Opens upward

0,0 0, = Opens downward

Directrix

Focus

Vertex0,

0,00,


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Example 2:1. Write the equation of the a parabola with

a)Vertex, 0,0 and focus, 3,0b)Vertex, 0,0and focus, 0, 5Hence, sketch each graph.

2. Find the focus and directrix of the parabola= 182 and sketch

the graph.


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3. Find the equation of a parabola that has vertex at origin, opensleft, and passes through the point 5,4.

4. Find the equation of the parabolas with following vertices and foci.a)Vertex 0,0; focus 0, 2b)Vertex

0,0

; focus

4,0


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The equation of a Parabola with Vertex , The standard equation for parabolas with vertex , is summarizedbelow.

Parabola Vertex Focus Directrix Shape

2 = 4

, + , = Opens tothe right

, , = + Opens tothe left

2 = 4 , , + =

Opensupward

, , = + Opens

downward


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5. State the vertex, focus and directrix for each of the followinga) 22 = 12 3b) 12 = 5+ 2

6. Sketch the graph of 22 = 12 1 showing clearly thefocus and directrix of the curve.


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Standard form and General form of parabolaThe equation 2 = 4 or 2 = 4 isknown as standard formof parabolic equation and it can be written inthe general formas

+ + + = or + + + = Finding vertex and focus of a parabola by completing the square

7. Write down the equation of given parabola below in standardform. For each parabola state the coordinates of the vertex, focus

and the equation of the directrix. Hence, sketch each graph.a) 2 + 8+ 4+ 12 = 0


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b) 2 + 8 2+ 22 = 0


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8. Find the equation of a parabola which satisfies the followingconditions, vertex 1, 2, its axis parallel to the -axis and theparabola passes through the point (3,6)


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Applications of parabola9. A necklace hanging between two fixed points and at the same

level. The length of the necklace between the two points is 100cm.

The midpoint of the necklace is 8cm below and . Assume thatthe necklace hangs in the form of parabolic curve, find theequation of the curve.


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10.A gigantic gate to the entrance of a theme park in the shape of aparabola is constructed on a level ground. The horizontal distance

between the end points to the gigantic gate is 20 meters. The

maximum height of the gigantic gate from the ground is 5meters.

Calculate the height of the gigantic gate at a horizontal distance of4meters from one of the end points.


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1 4EllipseDefinition of an EllipseAn ellipse is the set of point in a plane the sum of whose distances from

two fixed points 1and 2is a constant. (see figure 5)

Figure 5 Figure 6

In order to obtain the simplest equation for an ellipse, we place the foci

on the -axis at the point , 0and , 0as in Figure 6 so that theorigin is halfway between the foci. Let the sum of the distances from a

point on the ellipse to the foci be 2

> 0. Then

,

is a point on

the ellipse when

1 + 2= 2That is + 2 + 2 + 2 + 2 = 2

Or

2 +

2 = 2

+

2 +

2

Squaring both sides, we have

2 2+ 2 + 2 = 42 4+ 2 + 2 + 2 + 2+ 2 + 2Which simplifies to

+ 2 + 2 = 2 +


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We Square again:

22 + 2+ 2 + 2= 4 + 22+ 22Which becomes

2

2

2

+ 2

2

= 2

2

2

From triangle 12in Figure 6 we see that 2< 2, so < and,therefore, 2 2 > 0. For convenience, let 2 = 2 2. Then theequation of the ellipse becomes 22 + 22 = 22or, if both sidesare divided by 22,

2

2+ 2

2 = 1

Where 2 = 2 2 For >

Figure 7

01, 02, 0 1, 02, 0

40,

30,

vertexvertex

latus rectum latus rectum

focusfocus


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The equation of a Ellipses with Vertex 0,0The standard equations of ellipses with centre 0,0are as summarizedas below

Ellipse Major axis Center Foci Vertex Latusrectum

22+

22 = 1

Where

2 =

2

2

For >

Horizontal

= 00,0 , 0 , 0 22

22+

22 = 1

Where

2 = 2 2For >

Vertical

= 00,0 0, 0, 22

Example 3:1. Find an equation of the ellipse with vertices 4,0 and foci2,0.


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2. Find the equation of the ellipse with centre 0,0with the verticesat 3,0and = 2.

3. Sketch the ellipse 2

9 + 2

4 = 1


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4. Sketch the ellipse 92 + 162 = 144

5. Find the equation for the ellipse that has its centre at the originwith vertices 0,7and foci 0,2.


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6. Find the equation for the ellipse that has its centre at the originwith vertices 0,5and minor axis of length 3. Sketch theellipse.

7. Find the focus and equation of the ellipse with centre0, 0

and

vertices at 0, 4and length of minor axis is 4


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8. Sketch the ellipse with equation 29

+216

= 1

9. Find the centre, vertices, foci, major and minor axes and length of thelatus rectum for the ellipse

2

16+

2

25= 1


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The equation of a Ellipses with Vertex , The standard equations of ellipses with centre , are as summarizedas below

Ellipse Major axis Center Vertex Foci Latusrectum

22 +

22 = 1

Where

2 = 2 2For >

Horizontal

= , , , 22

22 +

22 = 1

Where

2 = 2 2For >

Vertical

= , , , 22

10. Find the centre and the foci of the ellipse +329

+12

4 = 1


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11. Write the equation of the ellipse that has vertices at 3, 5and7, 5and foci at 1, 5and 5, 5.

12. Sketch the ellipse with equation 1225

+216

= 1


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13. Find the equation of an ellipse with centre 3,1 and the majoraxis running parallel with the -axis. The length of the major axisis 10units and the minor axis is 6units Sketch the ellipse.

14. Find the equation of ellipse with vertices8,5

and

10,1

with

centre 8, .


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Finding the centre and foci of and ellipse by completing square15. Sketch the graph of the equation

162 + 92 + 64 18 71 = 0


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16. Sketch the graph of the equation42 + 92 8 36+ 4 = 0


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1 5HyperbolaDefinition of the HyperbolaA hyperbola is the set of all points in the plane, the difference of whose

distance from two fixed points 1 and 2 is a constant. (see Figure 8).These two fixed points are the foci of the hyperbola.

Notice that the definition of a hyperbola is similar to that of an ellipse;the only change is that the sum of distances has become a difference ofdistance.

In fact, the derivation of the equation of a hyperbola is also similar tothe one given earlier for an ellipse. The difference of distances is

1 2= 2, then the equation of the hyperbola is22

22 = 1

Where

2 =

2 +

2.

When we draw a hyperbola it is useful to first draw its asymptotes,

which are the dashed lines = and = .

Figure 8

,

2, 01, 0 0


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Figure 9

The equation of a Hyperbola with Center 0,0The standard equation for Hyperbola with Center 0,0 is summarizedbelow.

Ellipse Major axis Center Vertex Foci AsymptotesLatus

rectum

22

22 = 1

Where

2 =

2 +

2

Horizontal

= 00,0 , 0 , 0 =

22

22

22= 1

Where

2 =

2 +

2

Vertical

= 0, 0, 0, =

22

2 11 2

Focus

Asymptotes

Vertex

Latus RectumLatus Rectum

Focus


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Example 4:1.Determine the centre, vertices, and foci of the hyperbola given by

the equation

= Also, determine the equations of the

asymptotes of this hyperbola.

2.Graph the hyperbola 2

4 = . Find the vertices, foci and

equations of the asymptotic lines.


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3.Graph the hyperbola 2

9 = , Give the vertices, foci and

equations of asymptotic lines.

4.Find an equation of a hyperbola with centre at the origin, onevertex at 7,0and a focus at 12,0.


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The equation of a Hyperbola with Center , The standard equation for Hyperbola with Center , is summarizedbelow.

Ellipse Majoraxis Center Vertex Foci Asymptotes Latusrectum

22

22 = 1

Where

2 = 2 + 2= , , , =

22

22

22 = 1

Where

2 = 2 + 2

=

, , , = 22

5.Graph the equation22

36 12

25 = 1. Find the centre, vertices,foci and the equations of the asymptotes.


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6.Find the equation of a hyperbola with centre 1,1, vertex3,1and focus at 5,1.

7.Find the equation of the hyperbola with vertices at 1,6 and1,

2

and foci at

1,7

and

1,

3

.


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Finding the centre and foci of hyperbola by completing the square8.Sketch the curve represented by the equation

92 42 18+ 32 91 = 0.


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9.Sketch the curve represented by the equation42 162 + 8 128 316 = 0.


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1 6The Intersection of Straight Line and Conic SectionsType of Points of IntersectionsThe coordinates of the points of intersection of straight line and conicsection can be found by solving two equations simultaneously.

If the quadratic equation;a)2 4= 0

has one real root

The line and the curve meet at the

same points, . The line istangentto the curve at

b)2 4> 0has two real roots

The line and the curve intersect at

two different points, and .

c)2 4< 0has no real roots

The line does not intersect theconic section at all

Example 5:1.Find the intersections between the circle 2 + 2 = 24 and line

+ 2= 5.


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2.Find the coordinates of the points of intersection between the circle2 + 2 6 4+ 9 = 0and the line = 7

3.The line passing through the point 2 , 2 on the curve

2 = 4

and the point

2,0

intersects the curve once again at

the point . Find the coordinates of the pointin terms of .


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4.Find the equations of the tangents with gradient 2 to the ellipsewith equation 22 + 32 = 6, and find their points of contact.

5.Show that part of the line 3= + 5 is a chord of the circle

2

+ 2

6 2 15 = 0and find the length if this chord.


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6.Prove line + + = 0 touches the ellipse22 + 22 = 22, then 22 + 22 = 2.

7.Find the condition that the line = + shall touch thehyperbola

2

2 2

2= 1.


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1 7Parametric Representations of Conic SectionsDefinitionThe equations = and = are called the parametricequationsfor the curve defined by set of points , and the variable is called parameter.

Conic Section Cartesian Equations Parametric Equations

1 Circle 2 + 2 = 2 = cos = sin

2 Parabola 2 = 4 = 2= 2

3 Ellipse22+

22 = 1

= cos = sin

4 Hyperbola2

2 2

2 = 1= sec = tan

Conic Section Cartesian Equations Parametric Equations

1 Circle 2 + 2 = 2 = cos + = sin +

2 Parabola 2 = 4 = 2

+ = 2+

3 Ellipse 2

2 + 2

2 = 1= cos + = sin +

4 Hyperbola 2

2

2

2 = 1

= sec +

= tan +


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Example 6:1. Show that the curve whose parametric equation are

= 1 + cos and = sin represent a circle.

2. Describe and graph the curve represented by the parametricequations

= cos

,

= sin

0

2

.


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5. For each of the following curves, obtain its parametric equations.(a) 2 = 12(b) 22 = 9

6. Graph the plane curve given parametrically by = 8 cos ,y = 4sin , < < . Identify the curve by eliminatingthe parameter .


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7. Find parametric equations for the conic section with the givenequations 252 + 92 100+ 54 144 = 0.

8. Find the Cartesian equation for the curve with the parametricequations = 3 + 2 cos and = 2 + sin . Hence sketch thecurve.


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9. Find parametric equations for the conic section with the givenequations 2 162 10+ 32 7 = 0.

10.A curve is given the parametric equations = 23 cos + 2 and= 3 sin 3where is parameter. Show that the curve is anellipse. Hence, find the coordinates at the center and foci of thecurve.

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Chapter 1 Conic Sections

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