1 Solutions PART I. A. For each of the following, use the template for

Algebra II: Strand 7. Conic Sections; Topic 2. Locus Definitions of Conic Sections; Task 7.2.1

December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board.

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TASK 7.2.1: LOCUS DEFINITIONS OF CONIC SECTIONS Solutions PART I. A. For each of the following, use the template for finding points, P, in a plane such that

( ) ( )1 2length PF length PF+ is the constant described. The radius of the smallest circle represents

a length of 1 unit and the radius of each subsequent concentric circle increases by 1 unit. 1. Locate all the points, P, on the circles such that the sum of the distances from P to each of the

marked points is 8.

The required points are at the intersections of circles: 1 unit from F1, 7 units from F2 2 units from F1, 6 units from F2 3 units from F1, 5 units from F2 4 units from F1, 4 units from F2 5 units from F1, 3 units from F2 6 units from F1, 2 units from F2 7 units from F1, 1 units from F2

2. Locate all the points, P, on the circles such that the sum of the distances from P to each of the

marked points is 10. The required points are at the intersections of circles: 2 units from F1, 8 units from F2 3 units from F1, 7 units from F2 4 units from F1, 6 units from F2 5 units from F1, 5 units from F2 6 units from F1, 4 units from F2 7 units from F1, 3 units from F2 8 units from F1, 2 units from F2

3. Locate all the points, P, on the circles such that the sum of the distances from P to each of the

marked points is 12.



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The required points are at the intersections of circles: 3 units from F1, 9 units from F2 4 units from F1, 8 units from F2 5 units from F1, 7 units from F2 6 units from F1, 6 units from F2 7 units from F1, 5 units from F2 8 units from F1, 4 units from F2 9 units from F1, 3 units from F2

4. Each set of points in exercises 1-3 are on a conic section. Identify the name of the conic

section and provide a definition of this conic section based upon your findings in exercises 1-3. The conic section is an ellipse. Definition: An ellipse is the set of all points in a plane such that the sum of the distances from the foci is constant.

B. Use the template for finding points, P, in a plane such that ( ) ( )1 2

length PF length PF! is a

constant. The radius of the smallest circle represents a length of 1 unit and the radius of each subsequent concentric circle increases by 1 unit. 1. Locate all the points, P, on the circles such that the absolute value of the difference of the

distances from P to each of the marked points is 4.

The required points are at the intersections of circles: 1 unit from F1, 5 units from F2 2 units from F1, 6 units from F2 3 units from F1, 7 units from F2 4 units from F1, 8 units from F2 5 units from F1, 9 units from F2 6 units from F1, 10 units from F2 7 units from F1, 11 units from F2

2. Locate all the points, P, on the circles such that the absolute value of the difference of the distances from P to each of the marked points is 3.



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The required points are at the intersections of circles: 1.5 units from F1, 4.5 units from F2 2 units from F1, 5 units from F2 3 units from F1, 6 units from F2 4 units from F1, 7 units from F2 5 units from F1, 8 units from F2 6 units from F1, 9 units from F2

3. Locate all the points, P, on the circles such that the absolute value of the difference of the distances from P to each of the marked points is 2.

The required points are at the intersections of circles: 2 units from F1, 4 units from F2 3 units from F1, 5 units from F2 4 units from F1, 6 units from F2 5 units from F1, 7 units from F2 6 units from F1, 8 units from F2 7 units from F1, 9 units from F2 8 units from F1, 10 units from F2

4. Each set of points in exercises 1-3 are on a conic section. Identify the name of the conic

section and provide a definition of this conic section based upon your findings in exercises 1-3. The conic section is a hyperbola. Definition: A hyperbola is the set of all points in a plan such that the absolute value of the difference of the distances from any point on the hyperbola to two given points, called foci, is constant.

C. Use the template for finding points, P, in a plane, that are equidistant from a designated line and a given designated point. The radius of the smallest circle represents a length of 1 unit and the radius of each subsequent concentric circle increases by 1 unit. The adjacent lines on the template are also one unit apart. 1. Choose the line that is 2 units below the given point, and locate all points P that are

equidistant from the line and point.



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2. Choose the line that is 4 units below the given point, and locate all points P that are equidistant from the line and point.

3. Choose the line that is 2 units above the given point, and locate all points P that are equidistant from the line and point.



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4. Each set of points in exercises 1-3 are on a conic section. Identify the name of the conic section and provide a definition of this conic section based upon your findings in exercises 1-3. The conic section is a parabola. Definition: A parabola is the set of all points in a plane that are the same distance from a given point called the focus and a given line called the directrix.

PART II. A. Use the template and sketches of the conic section, which you identified as an ellipse, from PART I to answer the following. 1. Describe the symmetries (if any) that are present. There are two lines of symmetry. These

lines are perpendicular. 2. The terms major axis, minor axis, and foci are often used to describe an ellipse. Explain

precisely what you think these terms mean. The foci are the two fixed points from which we determine the ellipse. The major axis is the line segment that is collinear with the line of symmetry that passes through the foci of the ellipse. The endpoints of this segment are the points on the ellipse that are farthest from each other. The minor axis is the line segment that is collinear with the line of symmetry perpendicular to the line of symmetry containing the major axis. We can also think of the major axis and minor axis as those portions of the lines of symmetry that are in the interior of the ellipse, where we define the interior as the finite area enclosed by the ellipse. The endpoints of these axes are on the ellipse.

3. Draw the major axis and minor axis on one of your sketches from PART II. Which point should be called the center of the ellipse? Explain in at least three different ways why the center bisects the major axis and the minor axis. The intersection of the major and minor axes is the center of the ellipse. One way to see that the center is a point of bisection on the major and minor axes is to verify this with paper folding and realizing that the symmetry across each line induces the bisection (otherwise the figure would not be symmetric about the line in question). Another way to see this is to draw an isosceles triangle with base on the major axis



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and vertex at one of the endpoints of the minor axis. Since we know that the major and minor axes are perpendicular, we can use the Isosceles Triangle Theorem to conclude that the minor axis bisects the major axis. We then draw an isosceles triangle with base on the minor axis and vertex on one of the endpoints of the major axis and use the same argument to conclude that the minor axis is also bisected. Another way to see this (this has the ITT embedded in it) is to note that if we connect the endpoints of the axes to form a rhombus, then the major and minor axes are diagonals of this rhombus. We know that the diagonals of a rhombus are perpendicular and bisect each other.

4. A standard convention in labeling this conic section is as follows:

The distance from the center to the end of the major axis is usually labeled "a". The distance from the center to the end of the minor axis is usually labeled "b". The distance from the center to a focus is usually labeled "c". Locate, label, and measure "a", "b", and "c" on the ellipses that you drew in PART I.

Use patty paper to measure “b” with the circles on the template.

Record your results in the table provided.

Sum 8 Sum 10 Sum 12 a 4 5 6 b ≈ 2.6 4 ≈ 5.2 c 3 3 3

5. Is there a relationship between the parameters a, b, and/or c and the constant sum specified?

If so, explain how you determined this relationship. If not, explain why not. Yes, the constant sum is twice the value of the parameter a. That is, ( ) ( )1 2

2length PF length PF a+ = . This can

be determined by inspection of the table.



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6. Determine a relationship (if it exists) among the parameters a, b, and c. Explain your reasoning. Draw an isosceles triangle with vertices at the foci, and one endpoint of the minor axis. Justify that the length of each of the congruent sides of the triangle must be “a” (e.g. the segments are of equal length and the sum of their lengths must be 2a). Again, the minor axis bisects the base of this isosceles triangle (thus it is also a perpendicular bisector)—thus there are two right triangles formed. The relationship 2 2 2

a c b= + arises from applying the Pythagorean theorem to one of the right triangles formed.

7. On a coordinate plane we are given the foci (0,3) and (0,-3) and constant sum=8. Write an equation that any point P(x,y) on the conic section must satisfy. Simplify your equation to the

form 2 2

1x y

n p+ = where n and p are constants. Conclude whether there is a relationship

between your constants n and p and the parameters a, b, and/or c. If so, state it. The coordinates of F1 are ( 0 , 3 ), F2 are ( 0 , -3 ), and P are ( x , y).

( ) ( )22

1 3length PF x y= + ! and ( ) ( )22

2 3length PF x y= + +

( ) ( )1 2

8length PF length PF+ =

( ) ( )2 22 2

3 3 8x y x y+ ! + + + =

x

2 + y + 3( )2

= 8 ! x2 + y ! 3( )

2

x2 + y + 3( )

2!

"#$

%&

2

= 8 ' x2 + y ' 3( )

2!

"#$

%&

2

x2 + y + 3( )

2

= 64 '16 x2 + y ' 3( )

2

+ x2 + y ' 3( )

2

x2 + y

2 + 6y + 9 = 64 '16 x2 + y ' 3( )

2

+ x2 + y

2 ' 6y + 9

12y = 64 '16 x2 + y ' 3( )

2

12y ' 64 = '16 x2 + y ' 3( )

2

4 '3

4y = x

2 + y ' 3( )2



8

4 !3

4y

"

#$%

&'

2

= x2 + y ! 3( )

2"

#$%

&'

2

16 ! 6y +9

16y

2 = x2 + y ! 3( )

2

16 ! 6y +9

16y

2 = x2 + y

2 ! 6y + 9

7 = x2 + y

2 !9

16y

2

7 = x2 +

7

16y

2

1=x

2

7+

y2

16

By observation, p = a2 and n = b2

8. On a coordinate plane we are given the foci (0,c) and (0,-c) and constant sum=2a. Write an equation that any point P(x,y) on the conic section must satisfy. Simplify your equation to the

form 2 2

1x y

n p+ = where n and p are constants. Does your simplified equation support your

conclusion in exercise 7? Explain. The coordinates of F1 are ( 0 , c ), F2 are ( 0 , -c ), and P are ( x , y).

( ) ( )22

1length PF x y c= + ! and ( ) ( )22

2length PF x y c= + +

( ) ( )1 2

2length PF length PF a+ =

( ) ( )2 22 2

2x y c x y c a+ ! + + + =

( ) ( )2 22 22x y c a x y c+ + = ! + !

( ) ( )2 2

2 22 22x y c a x y c

! " ! "+ + = # + #$ % $ %& ' & '

( ) ( ) ( )2 2 22 2 2 24 4x y c a a x y c x y c+ ! = ! + + + + +

( )22 2 2 2 2 2 2 2

2 4 4 2x y cy c a a x y c x y cy c+ ! + = ! + + + + + +



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( )

( )

( ) ( )

22 2

22 2

22 22 2

4 4 4cy a a x y c

cy a a x y c

cy a a x y c

! ! = ! + +

+ = + +

" #+ = + +$ %& '

( )( )

( ) ( )

22 2 2 4 2 2

2 2 2 4 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 4

2 2 2 2 2 2 4 2 2

2 2 2 2 2 2 2 2

2 2

2 2 2

2

2 2

1

c y ca y a a x y c

c y ca y a a x a y ca y a c

c y a y a x a c a

a y c y a x a a c

a c y a x a a c

y x

a a c

+ + = + +

+ + = + + +

! ! = !

! + = !

! + = !

+ =!

since ( )2 2 2

a c b! = , a substitution yields 2 2

2 21

y x

a b+ = Yes, this supports the result in exercise 7.

9. From the template provided in PART I, can one generate any ellipse desired? Why or why not? Explain your reasoning. No, because this will only generate ellipses whose foci are 6 units apart. However, if we wanted to draw an ellipse whose foci were d units apart we could reassign

the radius of the smallest circle to 6

d and adjust the units throughout the template.



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B. Use the template and sketches from PART I which you identified as a hyperbola to answer the following. 1. Describe the symmetries (if any) that are present. There are two lines of symmetry. These

lines are perpendicular.

2. Which point should be called the center of the hyperbola? Justify your reasoning. The intersection of the lines of symmetry should be called the center of the hyperbola. This is the only point that remains fixed when reflect the figure across the lines of symmetry in succession.

3. The point on each branch of the hyperbola that is nearest the center is a vertex of the hyperbola. What do the vertices, center, and foci of a hyperbola have in common? The vertices, center, and foci are collinear and are all on one of the lines of symmetry.

4. The distance from the center to a vertex of the hyperbola is usually labeled “a” and the distance from the center to a focus is usually labeled “c”.

Measure “a” and “c” on each hyperbola you sketched in PART I. Record your results in the following table.

Difference 4

Difference 3 Difference 2

a 2 1.5 1 c 3 3 3

5. Is there a relationship between the parameters a and/or c and the constant difference

specified? If so, explain how you determined this relationship. If not, explain why not. By inspecting the table values, we may determine that the constant difference is twice the value



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of the parameter a. That is, ( ) ( )1 22length PF length PF a! = .

6. The general equation that naturally arises from the hyperbolas sketched is 2 2

2 2 21

x y

c a a

!+ =

!.

Typically, the general equation for hyperbolas with this orientation is given as 2 2

2 21

x y

b a

!+ = .

Draw a circle of radius c, centered at the center of the hyperbola. Find all points on this circle that illustrate the relationship 2 2 2

b c a= ! . Use your sketch to identify a segment of length 2a whose endpoints are on the vertices of the hyperbola and a segment of length 2b that is perpendicular to this segment. The former is called the transverse axis of the hyperbola and the latter is called the conjugate axis of the hyperbola. Explain why we can completely determine the equation of a hyperbola with transverse axis parallel to the y-axis if we are given the center and the lengths of the transverse and conjugate axes. What can we determine if we are not given that the transverse axis is parallel to the y-axis?

b= c2-a2

Length from center to i1 is c.

F2

(b,0)

i2i3

i1i4 (0,a)

(0,c)

(0,0)

From the form 2 2 2b c a= ! , notice that we are looking for right triangles with hypotenuse of length c and one leg of length a. A triangle with vertices at the center of the hyperbola, at one of the vertices of the hyperbola, and on the circle of radius c drawn will have side lengths a and c. To ensure that the side with length c is the hypotenuse of a right triangle, we draw the lines that pass through the vertices of the hyperbola and are perpendicular to the line of symmetry passing through the vertices of the hyperbola.

We can completely determine the equation of a hyperbola if we are given the lengths of the axes, the center, and that the transverse axis is parallel to the y-axis because the length of the transverse axis gives us the constant difference and the location of the foci can be determined



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by using the values of b and a to find c and placing the foci on a line parallel to the y-axis that passes through the center.

If we are not given the “slope” of the transverse axis, then we can only determine a family of hyperbolas with the given center and lengths of the transverse and conjugate axes. If we draw a hyperbola with transverse axis parallel to the y-axis, we can rotate this about the center of the hyperbola to produce the family of hyperbolas.

7. From the template provided in PART I, can one generate any hyperbola desired? Why or why not? Explain your reasoning. No, because this will only generate hyperbolas whose foci that are 6 units apart. However, if we wanted to draw a hyperbola whose foci were d units

apart we could reassign the radius of the smallest circle to 6

d and adjust the units throughout

the template.

C. Use the template and sketches of the conic section, which you identified as a parabola, from PART I to answer the following. 1. Describe the symmetries (if any) that are present. There is one line of symmetry.

2. The given point is called the focus of the parabola and the given line is called the directrix of

the parabola. The vertex of the parabola is the point on the parabola closest to the directrix. What do the vertex and the focus have in common? The vertex and the focus are on the line of symmetry.

3. Determine relationships (if any) between the vertex, focus, and directrix. Explain. The vertex is the midpoint between the focus and the point that is the intersection of the line of symmetry and the directrix. This can be deduced from the fact that the distance from the vertex to the focus must be the same as the distance from the focus to the directrix, or this follows from the locus definition of a parabola.

4. The distance from the vertex to the focus is usually labeled “a”. Use the distance formula to determine a general equation for a parabola with vertex at the origin and focus on the positive y-axis. The vertex has coordinates (0,0), the focus has coordinates (0,a), and the directrix is the line y=-a. A point P(x,y) on the plane that is on the parabola must satisfy the following relationship: distance from P(x,y) to (0,a)=distance from P(x,y) to directrix y=-a . We must recall that the distance from a point P to a line (the directrix in this case) is determined by the length of the line segment with endpoint P and perpendicular to the given line. Thus, we can rewrite the relationship as: distance from P(x,y) to (0,a)=distance from P(x,y) to (x,-a) . Now, we use the distance formula to write an equation that arises from this relationship:



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( ) ( ) ( ) ( )2 2 2 2

0x y a x x y a! + ! = ! + + . Simplifying:

( ) ( )

( ) ( )

2 22

2 22

2 2 2 2 2

2

2

2 2

4

4

x y a y a

x y a y a

x y ay a y ay a

x ay

xy

a

+ ! = +

+ ! = +

+ ! + = + +

=

=

5. How does the parameter “a” influence the shape of the parabola in exercise 4? Explain. We can relate this to the transformations earlier in the course to see that the value of a causes a vertical stretch or dilation of the parent function 2

y x= . For example, for small values of a we have a vertical stretch of the parent function 2

y x= .

6. We can use the distance formula to determine the general equation of a parabola with vertex at the origin and focus on the negative y-axis. Can we use the result in exercise 4 to determine this general equation in another way? Explain. Yes, we reflect this across the x-

axis. That is, if we replace points (x,y) with points (x,-y), we obtain 2

4

xy

a

!= .

Teaching notes Part I.

A. Guide the participants to complete the template by demonstrating on a transparency of the template how to locate the first set of points. Have them complete the rest of the exercises 1-4 in pairs.

B. Guide the participants to complete the template by demonstrating on a transparency of the template how to locate the first set of points. Have them complete the rest of the exercises 1-4 in pairs.

C. Guide the participants to complete the template by demonstrating on a transparency of the template how to locate the first set of points. Have them complete the rest of the exercises 1-4 in pairs.



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After all pairs have identified the conic sections and proposed a locus definition for each, form groups of four and have each group come to a consensus on the best definitions possible at this stage of the task. Have each group write their definitions on chart paper and post it in the classroom. Have the participants review the postings of the other groups. Clarify any discrepancies that may be present among their definitions.

Part II. Let participants work in small groups on Part II. Suggest that paper folding may help them determine any symmetries that may exist. A. In exercise 4, it may be helpful to demonstrate how to measure the length of “b” with patty paper. Demonstrate this using the transparency of the template:

Copy the length "b" onto patty paper, then turn it so that one end of the segment aligns with the center of one set of concentric circles. Its length can then be approximated as the radius of a circle.

In exercise 6, encourage participants to draw triangles using the center, foci, and endpoints of the major and minor axes, to determine a relationship among the parameters a, b, and c. In exercises 7 and 8, it is important to allow participants time to work through the calculations. For many, this will be the first time that they have ever derived the equation of an ellipse using the locus definition. After participants have had time to answer exercise 9, discuss their findings. Lead a discussion on how we may rotate the template 90º clockwise to consider ellipses with a horizontal major axis. Have them discuss what changes in the standard form of the equation will arise from this.

B. Allow participants time in their groups to discuss their observations and justifications in exercises 1-5. For exercise 6, explain that they can derive the equation given and that it follows from simplifying the locus definition. It is important to allow participants time to determine where the parameter “b” arises in relation to the other parameters. When participants have moved on to exercise 7 and completed it, lead a summary discussion on their finds in exercise 6. Follow the latter with a discussion on how we may rotate the template 90º clockwise to consider hyperbolas opening left and right. Lead a discussion of what changes in the standard form equation. C. Allow participants to work on 1-6 in groups. As they work in groups, the facilitator my have to remind them how to determine distance between a point and a line, the locus definition for a parabola, and transformation ideas from previous work to answer exercises 5 and 6.



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TASK 7.2.1: LOCUS DEFINITIONS OF CONIC SECTIONS PART I. A. For each of the following, use the template for finding points, P, in a plane such that

( ) ( )1 2length PF length PF+ is the constant described. The radius of the smallest circle represents

a length of 1 unit and the radius of each subsequent concentric circle increases by 1 unit. 1. Locate all the points, P, on the circles such that the sum of the distances from P to each of the

marked points is 8. 2. Locate all the points, P, on the circles such that the sum of the distances from P to each of the

marked points is 10. 3. Locate all the points, P, on the circles such that the sum of the distances from P to each of the

marked points is 12. 4. Each set of points in exercises 1-3 are on a conic section. Identify the name of the conic

section and provide a definition of this conic section based upon your findings in exercises 1-3.

B. Use the template for finding points, P, in a plane such that length PF

1( ) ! length PF

2( ) is a

constant. The radius of the smallest circle represents a length of 1 unit and the radius of each subsequent concentric circle increases by 1 unit. 1. Locate all the points, P, on the circles such that the absolute value of the difference of the

distances from P to each of the marked points is 4. 2. Locate all the points, P, on the circles such that the absolute value of the difference of the

distances from P to each of the marked points is 3. 3. Locate all the points, P, on the circles such that the absolute value of the difference of the

distances from P to each of the marked points is 2.

4. Each set of points in exercises 1-3 are on a conic section. Identify the name of the conic section and provide a definition of this conic section based upon your findings in exercises 1-3.

C. Use the template for finding points, P, in a plane that are equidistant from a designated line and a given designated point. The radius of the smallest circle represents a length of 1 unit and the radius of each subsequent concentric circle increases by 1 unit. The adjacent lines on the template are also one unit apart.



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3. Choose the line that is 2 units above the given point, and locate all points P that are equidistant from the line and point.

4. Each set of points in exercises 1-3 are on a conic section. Identify the name of the conic section and provide a definition of this conic section based upon your findings in exercises 1-3.



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18



19

Template for finding points, P, that are equidistant from point F to a designated line l.

F



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PART II. A. Use the template and sketches of the conic section, which you identified as an ellipse, from PART I to answer the following. 1. Describe the symmetries (if any) that are present. 2. The terms major axis, minor axis, and foci are often used to describe an ellipse. Explain

precisely what you think these terms mean. 3. Draw the major axis and minor axis on one of your sketches from PART II. Which point

should be called the center of the ellipse? Explain in at least three different ways why the center bisects the major axis and the minor axis.

4. A standard convention in labeling this conic section is as follows:

The distance from the center to the end of the major axis is usually labeled "a". The distance from the center to the end of the minor axis is usually labeled "b". The distance from the center to a focus is usually labeled "c". Locate, label, and measure "a", "b", and "c" on the ellipses that you drew in PART I. Record your results in the table provided.

Sum 8 Sum 10 Sum 12 a b c

5. Is there a relationship between the parameters a, b, and/or c and the constant sum specified?

If so, explain how you determined this relationship. If not, explain why not.

6. Determine a relationship (if it exists) among the parameters a, b, and c. Explain your reasoning.

7. On a coordinate plane we are given the foci (0,3) and (0,-3) and constant sum=8. Write an equation that any point P(x,y) on the conic section must satisfy. Simplify your equation to the

form

x2

n+

y2

p= 1 where n and p are constants. Conclude whether there is a relationship

between your constants n and p and the parameters a, b, and/or c. If so, state it.

8. On a coordinate plane we are given the foci (0,c) and (0,-c) and constant sum=2a. Write an equation that any point P(x,y) on the conic section must satisfy. Simplify your equation to the



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form 2 2

1x y

n p+ = where n and p are constants. Does your simplified equation support your

conclusion in exercise 7? Explain.

9. From the template provided in PART I, can one generate any ellipse desired? Why or why not? Explain your reasoning.



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B. Use the template and sketches of the conic section from PART I, which you identified as a hyperbola, to answer the following. 1. Describe the symmetries (if any) that are present.

2. Which point should be called the center of the hyperbola? Justify your reasoning.

3. The point on each branch of the hyperbola that is nearest the center is a vertex of the

hyperbola. What do the vertices, center, and foci of a hyperbola have in common?

4. The distance from the center to a vertex of the hyperbola is usually labeled “a” and the distance from the center to a focus is usually labeled “c”. Measure “a” and “c” on each hyperbola you sketched in PART I. Record your results in the following table.

Difference 4 Difference 3 Difference 2

a c

5. Is there a relationship between the parameters a and/or c and the constant difference

specified? If so, explain how you determined this relationship. If not, explain why not.

6. The general equation that naturally arises from the hyperbolas sketched is

!x2

c2

! a2

+y

2

a2

= 1. Typically, the general equation for hyperbolas with this orientation is

given as

!x2

b2

+y

2

a2

= 1. Draw a circle of radius c, centered at the center of the hyperbola.

Find all points on this circle that illustrate the relationship b2

= c2

! a2 . Use your sketch to

identify a segment of length 2a whose endpoints are on the vertices of the hyperbola and a segment of length 2b that is perpendicular to this segment. The former is called the transverse axis of the hyperbola and the latter is called the conjugate axis of the hyperbola. Explain why we can completely determine the equation of a hyperbola with transverse axis parallel to the y-axis if we are given the center and the lengths of the transverse and conjugate axes. What can we determine if we are not given that the transverse axis is parallel to the y-axis?

7. From the template provided in PART I, can one generate any hyperbola desired? Why or why not? Explain your reasoning.

C. Use the template and sketches from PART I which you identified as a parabola to answer the following.



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1. Describe the symmetries (if any) that are present.

2. The given point is called the focus of the parabola and the given line is called the directrix of the parabola. The vertex of the parabola is the point on the parabola closest to the directrix. What do the vertex and the focus have in common?

3. Determine relationships (if any) between the vertex, focus, and directrix. Explain.

4. The distance from the vertex to the focus is usually labeled “a”. Use the distance formula to determine a general equation for a parabola with vertex at the origin and focus on the positive y-axis.

5. How does the parameter “a” influence the shape of the parabola in exercise 4? Explain.

6. We can use the distance formula to determine the general equation of a parabola with vertex at the origin and focus on the negative y-axis. Can we use the result in exercise 4 to determine this general equation in another way? Explain.

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1 Solutions PART I. A. For each of the following, use the template for