• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
Page
  1. 1
    1
  2. 2
    2
  3. 3
    3
  4. 4
    4
  5. 5
    5
  6. 6
    6
  7. 7
    7
  8. 8
    8
  9. 9
    9
  10. 10
    10
  11. 11
    11
  12. 12
    12
  13. 13
    13
  14. 14
    14
  15. 15
    15
  16. 16
    16
  17. 17
    17
  18. 18
    18
  19. 19
    19
  20. 20
    20
  21. 21
    21
  22. 22
    22
  23. 23
    23
  24. 24
    24
  25. 25
    25
  26. 26
    26

Mathematics SL portoflio type 1(circles)

Extracts from this document...

Introduction

Background information:

The general form of a circle is denoted by  image00.png

 where r is the radius of the circle, k stands for the vertical translation and h for the horizontal translation of the circle on the coordinate plane. We derive the equation of the circle by transforming the general formula to get image01.png

 , where + sign corresponds to the upper half of the circle and the – sign corresponds to the bottom half of the circle. This equation will be used throughout the investigation to assist with algebraic problems concerning the relationship between r, OP and OP’.  For the purpose of this investigation, O will correspond to (0,0) on the coordinate plane.

Part 1

The purpose of this exercise is to determine the relationship between OP and OP’ with constant r of 1. However, it is important to note that r only implies to the radius of circles C1 and C3.

Equation of circle C2

image02.png

image03.png

image04.png

Equation of circle C1

image05.png

image03.png

image06.png

Centre of circle C3 is the upper intersection of circles C1 and C2 , thus we can find the middle of circle C3 by determining the value of x and y where both formula of the circles C1 and C2  are equal to each other, as shown below. We take the positive value of the root of the equation, as we are interested in the upper intersection of the two circles, which corresponds to the circle C1.

C2=C1

image07.png

image08.png

image09.png

image10.png

image11.png

image12.png

 ; image13.png

image14.png

image15.png

image16.png

image17.png

image18.png

The coordinate of midpoint of circle C3 is  

image19.png

...read more.

Middle

image77.png

image78.png

image79.png

image80.png

image81.png

image82.png

image106.png

Similarly, the same technique was used to find OP’ with different values of r and constant value of OP of 2

The results have been displayed in the table below:

r

2

3

4

OP’

2

4.5

8

image83.png

 When the value of OP is kept constant at 2, OP’ is equivalent to r2 divided by 2.

image84.png

However, there are some limitations to the equation above. By studying the diagram below we can see that as the radius increases, the value of OP’ also increases. My statement is supported by the data in the table above and on the observations of the diagrams below. However, if the value of r is beyond a certain point, there will be no intersection between circles C1 and C2, thus giving no value point A and hence OP’ (the circle C3 will not exist). The maximum value of r (with constant OP of 2) which gives the solution for OP’ is 4. Radius above 4 will lead to the circle C2 to be inside the circle C1 hence giving no intersection between the 2 circles and no solution for OP’.

image117.png

On the diagram below, when can clearly see the situation when C2 is inside C1, with not intersection of the two circles, leading to the non-existence of C3 and giving no solution for OP’

From the diagrams below, we can see that for constant OP of two, increasing the radius casues the value of OP’ to increase ,untill r reaches its maximum to give a solution .

image123.pngimage124.png

image125.png

Maximum value of OP (4)

...read more.

Conclusion

1 and C2, hence, point A will not exist and there is no solution for OP’. However, not always when r is bigger than OP, there is no solution for OP’. This has been further investigated in Part 2 using graphing technology.

Conclusion

Throughout the investigating we have found general statement which link all three variables: OP (radius of C2 or distance between O and C2) , r (radius of C1 and C3) and OP’(the intersection of C3 with the x-axis or OP.  The formula is represented by:

image98.png

Through part A, we have found the relation between OP and OP’, with a constant r. In part B we have found the relation of OP’ and r, with constant OP and in C we have found the combined general formula by using the data generated in part A and B, and manipulating all variables r, OP and OP’.

Limitations of the general formula were determined by the use of technology, which would give no values of OP’,(through the non-existence of circle C3). r could not be greater than OP as this would result in no intersection between C1 and C2 and would not lead to solution for OP’. For this reason, r must be smaller or equal to OP for there to be a value of OP’ and for the combined general statement to exist.

Appendix A (for part 3)image115.png

image102.png

image116.png

image103.png

image103.png

image118.png

                                                                               OP = 2           r=1         OP’=0.5

image119.png

When OP=3; r=1 then OP’= 1/3

image120.png

image104.png

image121.png

When OP=-2; r=1 then OP’=-1/2

Due to inaccuracy of the results obtained using graphing technology, we assume that the P’ is (- 1/3, 0)image122.png

When OP=-3; r=1 then OP’=- 1/3

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Math IA type 2. In this task I will be investigating Probabilities and investigating ...

    Odds of Player C winning = Then I put this equation for the function for column 5 and got my results for the various values of c. From the values in table 4, I observe that as the values of c increase, the winning probabilities get close together and the

  2. Moss's Egg. Task -1- Find the area of the shaded region inside the two ...

    What we don't know however, is the triangle's perpendicular height, and we must refer to the dynamics of the triangle to determine this. It is noticed that chords AC and BC both meet at the same location, the point which links the radius of the small circle (C)

  1. Maths Investigation: Pascals Triangles

    The height of the triangle is also two rows. On shading multiples of 5, we have an inverted triangle which again is symmetrical around the middle on a base of four numbers. This pattern is then repeated but this time on two sides of the pyramid, and the base is again separated by one number (252).

  2. Math Portfolio: trigonometry investigation (circle trig)

    equals a positive number on the positive x axis and the value of r equals a positive number as mentioned beforehand. Therefore, when the value of x is divided by the value of r, a positive number is divided by a positive number resulting to a positive number.

  1. Math IA Type 1 Circles. The aim of this task is to investigate ...

    When Excel is used, five significant figures will be accepted because this is the standard number of significant figures that Excel presents. Using a constant number of significant digits allows for the same degree of accuracy through out the whole investigation and keeps the investigation orderly.

  2. MATH IA: investigate the position of points in intersecting circles

    This can be seen as: Get rid of the radicals. Add x2 to both sides Solve for x OP?=2GP= When OP=3 When OP=3 we will use the same method used to find OP? when OP=2. AP?=OA ?OAP is an isosceles triangle.

  1. Mathematic SL IA -Circles (scored 17 out of 20)

    a2= b2+c2-2bcCosA Problem Let r=1, find OP’ when OP=2, OP=3, and OP=4. * When OP=2, Draw a perpendicular line, âµ ∴ âµ ∴ When = âµ âµ ∴ Link AP’, âµp’ is on C3, ∴ , ∴ is an isosceles triangle.

  2. Maths IA. In this task I am asked to investigate the positions of ...

    Side?s A to O and A to P? are both 1, because they are the radiuses of point A (midpoint of triangle C3), I am told that r=1(r being the radius). In the first part of this problem I have to find the distance from O to P?, considering that O to P = 2.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work