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Coordinate Systems

Coordinate Systems. Introduction. In this chapter you will learn how to solve problems in coordinate systems You will learn what parametric equations are (if you have done C4 you will already have seen this!) You will learn the general equation of a parabola and what a parabola is

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Coordinate Systems

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  1. Coordinate Systems

  2. Introduction • In this chapter you will learn how to solve problems in coordinate systems • You will learn what parametric equations are (if you have done C4 you will already have seen this!) • You will learn the general equation of a parabola and what a parabola is • You will learn about the directrix and focus of a parabola • You will learn the equation for a rectangular hyperbola • You will learn to solve problems relating to the tangent and normal at points on a parabola and/or hyperbola • You will also practice algebraic methods in these types of question

  3. Teachings for exercise 3A

  4. Coordinate Systems You need to know what parametric equations are Parametric equations are where the x and y coordinates of each point along a curve are written as functions of a third parameter (usually ‘t’) So: x = f(t) y = g(t) Coordinates on a curve can be defined using parametric equations Sketch the curve given by the following parametric equations: As with plotting any curve, you can start by finding a set of values  Choose values for t and calculate both x and y in these cases… t - 3 - 2 - 1 0 1 2 3 x = t 9 4 1 0 1 4 9 2 y = 4t - 12 - 8 - 4 0 4 8 12 Now you have a set of coordinates, you can plot them on a standard Cartesian axis 3A

  5. Coordinate Systems You need to know what parametric equations are Parametric equations are where the x and y coordinates of each point along a curve are written as functions of a third parameter (usually ‘t’) So: x = f(t) y = g(t) Coordinates on a curve can be defined using parametric equations t 0 1 2 3 9 4 1 0 1 4 9 y = 4t 0 4 8 12 y 10 - 3 - 2 - 1 x = t 2 - 12 - 8 - 4 x 10 -10 3A

  6. Coordinate Systems You need to know what parametric equations are A curve has parametric equations: Find the Cartesian equation of the curve • A Cartesian version of the equation will be written in terms of x and y only • It will usually have the y and x terms on separate sides of the equals sign… • To do this you will need to rewrite one of the equations in terms of t and substitute it into the other… Divide by 2a Now you can replace t in the other equation… Replace t Square the fraction Combine terms ‘Cancel’ an ‘a’ term from the top and bottom This is the general equation for a parabola! Multiply by 4a 3A

  7. Coordinate Systems You need to know what parametric equations are A curve has parametric equations: Find the Cartesian equation of the curve, and hence, sketch it… • The previous method will work here as well, but there is an alternative way you can combine the equations – you will have to use your judgement as to whether this will always work! • The top of the fraction will be positive, and hence the curve will take the reciprocal shape (you should recognise this from Core 1 and GCSE!) Multiplying the equations together will actually cancel the ‘t’ terms straight away! Combine the right hand side ‘Cancel’ the ‘t’ terms Divide by x 3A

  8. Teachings for exercise 3B

  9. Coordinate Systems You need to know the general equation of a parabola, as well as its focus-directrix properties To the right is an example of a parabola with parametric equations: It will have a Cartesian equation in the form: where a is a positive constant. The general coordinate on the curve, P, can be expressed in a Cartesian or Parametric way… “A Parabola is the locus of points that are the same distance from a fixed point known as the focus, and a fixed line known as the directrix” y (0,0) x 3B

  10.  Parametric Form Cartesian Form  Coordinate Systems You need to know the general equation of a parabola, as well as its focus-directrix properties “A Parabola is the locus of points that are the same distance from a fixed point known as the focus, and a fixed line known as the directrix” The coordinates of the focus are given by: The equation of the directrix is given by: or y Directrix Focus x 3B

  11.  Parametric Form Cartesian Form  Coordinate Systems  Directrix  Focus You need to know the general equation of a parabola, as well as its focus-directrix properties Find the equation of the parabola with focus (7,0) and directrix x + 7 = 0 • If you compare these to the general form, you can see that the value of ‘a’ is 7 • So we replace ‘a’ with 7 in the Cartesian form of the equation… Substitute a = 7 3B

  12.  Parametric Form Cartesian Form  Coordinate Systems  Directrix  Focus You need to know the general equation of a parabola, as well as its focus-directrix properties Find the equation of the parabola with focus (√3/4,0) and directrix x = -√3/4 • If you compare these to the general form, you can see that the value of ‘a’ is √3/4 • So we replace ‘a’ with √3/4 in the Cartesian form of the equation… Substitute a = √3/4 3B

  13.  Parametric Form Cartesian Form  Coordinate Systems  Directrix  Focus You need to know the general equation of a parabola, as well as its focus-directrix properties Find the coordinates of the focus and an equation of the directrix for the parabola with equation:  Divide the coefficient of x by 4 to find the value of a Focus Directrix Substitute a = √2 = = = 3B

  14.  Parametric Form Cartesian Form  Coordinate Systems  Directrix  Focus y You need to know the general equation of a parabola, as well as its focus-directrix properties A point P(x,y) obeys a rule such that the distance of P to the point (6,0) is the same as the distance from P to the straight line x + 6 = 0. Prove that the locus of P has an equation of the form y2 = 4ax, and find the value of a. • We need to find ways to express the two distances, and then equate them… • The distance PD is just the sum of x and 6 P(x,y) D F 6 x x (6,0) x + 6 = 0 3B

  15.  Parametric Form Cartesian Form  Coordinate Systems  Directrix  Focus y You need to know the general equation of a parabola, as well as its focus-directrix properties A point P(x,y) obeys a rule such that the distance of P to the point (6,0) is the same as the distance from P to the straight line x + 6 = 0. Prove that the locus of P has an equation of the form y2 = 4ax, and find the value of a. • We need to find ways to express the two distances, and then equate them… • The distance PF can be found using Pythagoras’ Theorem and a bit of Algebra! P(x,y) D y F x 6 - x (6,0) x + 6 = 0 3B

  16.  Parametric Form Cartesian Form  Coordinate Systems  Directrix  Focus y You need to know the general equation of a parabola, as well as its focus-directrix properties A point P(x,y) obeys a rule such that the distance of P to the point (6,0) is the same as the distance from P to the straight line x + 6 = 0. Prove that the locus of P has an equation of the form y2 = 4ax, and find the value of a. P(x,y) D y F x 6 - x (6,0) Square both sides Sub in values x + 6 = 0 Multiply out brackets Subtract 36, subtract x2 Add 12x The equation is y2 = 24x and the value of a = 6 3B

  17. Teachings for exercise 3C

  18.  Parametric Form Cartesian Form  Coordinate Systems Directrix Focus  You need to be able to combine other aspects of coordinate geometry with parabolas The point P(8,-8) lies on the parabola C with equation y2 = 8x. The point S is the focus of the parabola. The straight line l passes through S and P. • Find the coordinates of S • Find an equation for l, giving your answer in the form ax + by + c = 0, where a, b and c are integers. Divide the coefficient of x by 4 to find ‘a’ This allows us to write the coordinates of the focus Focus  Line l passes through (2,0) and (8,-8)  You can use one of the linear curve equations from C1! Sub in the coordinates (carefully!) Simplify Cross-multiply Move terms so it is in the correct form Divide all by 2 to simplify 3C

  19.  Parametric Form Cartesian Form  Coordinate Systems Directrix Focus  To find where the line meets the parabola, you need to find a way to combine the equations (this will not always be the same way!) You need to be able to combine other aspects of coordinate geometry with parabolas The point P(8,-8) lies on the parabola C with equation y2 = 8x. The point S is the focus of the parabola. The straight line l passes through S and P. • Find the coordinates of S • Find an equation for l, giving your answer in the form ax + by + c = 0, where a, b and c are integers. The line l meets the parabola again at point Q c) Find the coordinates of Q d) Find the midpoint of PQ e) Sketch the parabola C, the line l and the points P, Q, S and M on the same diagram Double In terms of x Replace 8x with the equivalent expression in y Set equal to 0 Factorise Solve We already know the intersection where y = -8  Sub in y = 2 to find the other intersection! 3C

  20.  Parametric Form Cartesian Form  Coordinate Systems Directrix Focus  To find the midpoint of PQ you just find the mean of the x and y-coordinates… You need to be able to combine other aspects of coordinate geometry with parabolas The point P(8,-8) lies on the parabola C with equation y2 = 8x. The point S is the focus of the parabola. The straight line l passes through S and P. • Find the coordinates of S • Find an equation for l, giving your answer in the form ax + by + c = 0, where a, b and c are integers. The line l meets the parabola again at point Q c) Find the coordinates of Q d) Find the midpoint of PQ e) Sketch the parabola C, the line l and the points P, Q, S and M on the same diagram Sub in values Calculate Or you can use fractions! 3C

  21.  Parametric Form Cartesian Form  Coordinate Systems Directrix Focus  You need to be able to combine other aspects of coordinate geometry with parabolas The point P(8,-8) lies on the parabola C with equation y2 = 8x. The point S is the focus of the parabola. The straight line l passes through S and P. • Find the coordinates of S • Find an equation for l, giving your answer in the form ax + by + c = 0, where a, b and c are integers. The line l meets the parabola again at point Q c) Find the coordinates of Q d) Find the midpoint of PQ e) Sketch the parabola C, the line l and the points P, Q, S and M on the same diagram l C y Q(0.5,2) S(2,0) x M(4.25,-3) P(8,-8) 3C

  22. Teachings for exercise 3D

  23. Coordinate Systems You need to know an equation for a rectangular hyperbola and be able to find the equations of tangents and normals The reciprocal function you have seen before is also known as a hyperbola • The general shape can be thought of as being created when a double cone is sliced vertically (see right) • A hyperbola will always have 2 asymptotes • A rectangular hyperbola is one where the asymptotes are perpendicular to each other… Double cone being sliced vertically Asymptotes 3D

  24. Coordinate Systems You need to know an equation for a rectangular hyperbola and be able to find the equations of tangents and normals The curve shown is an example of a rectangular hyperbola with parametric equations: where c is a positive constant. • The Cartesian version of this equation is: • A general point on the curve has coordinates: 3D

  25. Coordinate Systems You need to know an equation for a rectangular hyperbola and be able to find the equations of tangents and normals The point P, where x = 2, lies on the rectangular hyperbola H with equation xy = 8. Find the equations of the tangent and normal to H at P, giving your answers in the form ax + by + c = 0, where a, b and c are integers. • You will need to use differentiation from C1/C2 to find the gradient at P Divide by x Write in index form Differentiate Rewrite so it is easier to sub in values At P, x = 2 Calculate 3D

  26. Coordinate Systems You need to know an equation for a rectangular hyperbola and be able to find the equations of tangents and normals The point P, where x = 2, lies on the rectangular hyperbola H with equation xy = 8. Find the equations of the tangent and normal to H at P, giving your answers in the form ax + by + c = 0, where a, b and c are integers. • You will need to use differentiation from C1/C2 to find the gradient at P • We also need a coordinate on the line to be able to work out its equation • Work out the y-coordinate at P Sub in x = 2 Calculate 3D

  27. Coordinate Systems You need to know an equation for a rectangular hyperbola and be able to find the equations of tangents and normals The point P, where x = 2, lies on the rectangular hyperbola H with equation xy = 8. Find the equations of the tangent and normal to H at P, giving your answers in the form ax + by + c = 0, where a, b and c are integers. • You will need to use differentiation from C1/C2 to find the gradient at P • We also need a coordinate on the line to be able to work out its equation • Work out the y-coordinate at P And use a formula from C1… Sub in the gradient and coordinate Multiply out the bracket Add 2x and subtract 4 to get the required form 3D

  28. Coordinate Systems Tangent You need to know an equation for a rectangular hyperbola and be able to find the equations of tangents and normals The point P, where x = 2, lies on the rectangular hyperbola H with equation xy = 8. Find the equations of the tangent and normal to H at P, giving your answers in the form ax + by + c = 0, where a, b and c are integers. • You will need to use differentiation from C1/C2 to find the gradient at P • We also need a coordinate on the line to be able to work out its equation • Work out the y-coordinate at P The normal passes through the same coordinate, but the gradient will be different (as in C1). Use -1/m Normal Sub in the gradient and coordinate Multiply out the bracket Double terms to remove the fraction Subtract 2y and add 8 to get the required form 3D

  29. Coordinate Systems 2x + y – 8 = 0 You need to know an equation for a rectangular hyperbola and be able to find the equations of tangents and normals The point P, where x = 2, lies on the rectangular hyperbola H with equation xy = 8. Find the equations of the tangent and normal to H at P, giving your answers in the form ax + by + c = 0, where a, b and c are integers. • This is what we worked out! P(2,4) Normal Tangent x - 2y + 6 = 0 Tangent  Normal  3D

  30. Coordinate Systems You need to know an equation for a rectangular hyperbola and be able to find the equations of tangents and normals The two distinct points, A and B, where x = 3, lie on the parabola C with equation y2 = 27x. The line l1 is the tangent to C at A and the line l2 is the tangent to C at B. Given that at A, y > 0: • Find the coordinates of A and B • Draw a sketch showing the parabola C, the points A and B and the lines l1 and l2 • Find equations for l1 and l2, giving your answers in the form ax + by + c = 0 where a, b and c are integers Sub in x = 3 Calculate Square root A must have the y-coordinate 9 as at A, y > 0 3D

  31. Coordinate Systems You need to know an equation for a rectangular hyperbola and be able to find the equations of tangents and normals The two distinct points, A and B, where x = 3, lie on the parabola C with equation y2 = 27x. The line l1 is the tangent to C at A and the line l2 is the tangent to C at B. Given that at A, y > 0: • Find the coordinates of A and B • Draw a sketch showing the parabola C, the points A and B and the lines l1 and l2 • Find equations for l1 and l2, giving your answers in the form ax + by + c = 0 where a, b and c are integers C y A(3,9) l1 x l2 B(3,-9) 3D

  32. Coordinate Systems You need to know an equation for a rectangular hyperbola and be able to find the equations of tangents and normals The two distinct points, A and B, where x = 3, lie on the parabola C with equation y2 = 27x. The line l1 is the tangent to C at A and the line l2 is the tangent to C at B. Given that at A, y > 0: • Find the coordinates of A and B • Draw a sketch showing the parabola C, the points A and B and the lines l1 and l2 • Find equations for l1 and l2, giving your answers in the form ax + by + c = 0 where a, b and c are integers  To find an equation for l1, we need the gradient at A (we already know the coordinate) Square root Split the root up Split the 27 up Simplify and rewrite √x using indices Now we will need to differentiate to find the gradient function 3D

  33. Coordinate Systems You need to know an equation for a rectangular hyperbola and be able to find the equations of tangents and normals The two distinct points, A and B, where x = 3, lie on the parabola C with equation y2 = 27x. The line l1 is the tangent to C at A and the line l2 is the tangent to C at B. Given that at A, y > 0: • Find the coordinates of A and B • Draw a sketch showing the parabola C, the points A and B and the lines l1 and l2 • Find equations for l1 and l2, giving your answers in the form ax + by + c = 0 where a, b and c are integers  To find an equation for l1, we need the gradient at A (we already know the coordinate) Differentiate Split terms Rewrite the x term Recombine We will get 2 values when we substitute in x.  These correspond to the different gradients of the tangents at A and B respectively. 3D

  34. Coordinate Systems You need to know an equation for a rectangular hyperbola and be able to find the equations of tangents and normals The two distinct points, A and B, where x = 3, lie on the parabola C with equation y2 = 27x. The line l1 is the tangent to C at A and the line l2 is the tangent to C at B. Given that at A, y > 0: • Find the coordinates of A and B • Draw a sketch showing the parabola C, the points A and B and the lines l1 and l2 • Find equations for l1 and l2, giving your answers in the form ax + by + c = 0 where a, b and c are integers  To find an equation for l1, we need the gradient at A (we already know the coordinate) Sub in x = 3 Cancel the √3 terms y A(3,9) C The tangent at A has a positive gradient so is 3/2 The tangent at B has a negative gradient so is -3/2 l1 x l2 B(3,-9) 3D

  35. Coordinate Systems You need to know an equation for a rectangular hyperbola and be able to find the equations of tangents and normals The two distinct points, A and B, where x = 3, lie on the parabola C with equation y2 = 27x. The line l1 is the tangent to C at A and the line l2 is the tangent to C at B. Given that at A, y > 0: • Find the coordinates of A and B • Draw a sketch showing the parabola C, the points A and B and the lines l1 and l2 • Find equations for l1 and l2, giving your answers in the form ax + by + c = 0 where a, b and c are integers Equation of line l1 Sub in the gradient and coordinate Multiply out the bracket Multiply all terms by 2 Subtract 2y and add 18 to get the required form 3D

  36. Coordinate Systems You need to know an equation for a rectangular hyperbola and be able to find the equations of tangents and normals The two distinct points, A and B, where x = 3, lie on the parabola C with equation y2 = 27x. The line l1 is the tangent to C at A and the line l2 is the tangent to C at B. Given that at A, y > 0: • Find the coordinates of A and B • Draw a sketch showing the parabola C, the points A and B and the lines l1 and l2 • Find equations for l1 and l2, giving your answers in the form ax + by + c = 0 where a, b and c are integers Equation of line l2 Sub in the gradient and coordinate Multiply out the bracket Multiply all terms by 2 Add 3x and subtract 9 to get the required form Key point: If you are only asked for one equation of a line, consider where it is on the parabola and whether the gradient will be positive or negative! 3D

  37. Teachings for exercise 3E

  38. Coordinate Systems You need to be able to find equations for tangents and normals algebraically The point P(at2,2at) lies on the parabola C with equation y2 = 4ax where a is a positive constant. Show that an equation of the normal to C at P is given by: • Start in the same way as before • We already have a coordinate P, we need an expression for the gradient • Differentiate the function… Square root both sides Root each part separately Rewrite the x term for differentiation Differentiate (remember ‘a’ is just a number so you don’t differentiate it as well!) Imagine the terms were separate… Rewrite the x term Combine This is our formula for the gradient at any given point on the parabola! 3E

  39. Coordinate Systems You need to be able to find equations for tangents and normals algebraically The point P(at2,2at) lies on the parabola C with equation y2 = 4ax where a is a positive constant. Show that an equation of the normal to C at P is given by: • Start in the same way as before • We already have a coordinate P, we need an expression for the gradient • Differentiate the function… • Now we can sub in the x-coordinate to find an expression for the gradient at P Replace x with the coordinate at P Split up terms Cancel the √a terms Square root the denominator So the gradient of the tangent at P = 1/t So the gradient of the normal at P = -t 3E

  40. Coordinate Systems You need to be able to find equations for tangents and normals algebraically The point P(at2,2at) lies on the parabola C with equation y2 = 4ax where a is a positive constant. Show that an equation of the normal to C at P is given by: • Start in the same way as before • We already have a coordinate P, we need an expression for the gradient • Differentiate the function… • Now we can sub in the x-coordinate to find an expression for the gradient at P Sub these into the equation of a line formula (as you would do if they were numerical) Sub in the gradient and the coordinate Multiply out the bracket Add 2at and add tx to get the required form Make sure you read the question carefully and check whether you’re finding the tangent or the normal! 3E

  41. Coordinate Systems You need to be able to find equations for tangents and normals algebraically The point P(ct,c/t), t ≠ 0, lies on the rectangular hyperbola H with equation xy = c2 where c is a positive constant. Show that an equation of the tangent to H at P is: • This will be similar to the last question, only with a hyperbola instead of a parabola • Find an expression for the gradient at a point on the curve, by differentiating Divide by x Write in a differentiatable form Differentiate – remember c2 is just a number Rewrite x-2 Combine 3E

  42. Coordinate Systems You need to be able to find equations for tangents and normals algebraically The point P(ct,c/t), t ≠ 0, lies on the rectangular hyperbola H with equation xy = c2 where c is a positive constant. Show that an equation of the tangent to H at P is: • This will be similar to the last question, only with a hyperbola instead of a parabola • Find an expression for the gradient at a point on the curve, by differentiating Sub in the x-coordinate at P Square the denominator Cancel the c2 terms So the gradient of the tangent to H at P is given by: 3E

  43. Coordinate Systems You need to be able to find equations for tangents and normals algebraically The point P(ct,c/t), t ≠ 0, lies on the rectangular hyperbola H with equation xy = c2 where c is a positive constant. Show that an equation of the tangent to H at P is: • This will be similar to the last question, only with a hyperbola instead of a parabola • Find an expression for the gradient at a point on the curve, by differentiating Sub these into the equation of a line formula (as you would do if they were numerical) Sub in the coordinate and the gradient Multiply out the bracket Multiply all terms by t2 (ensure you do this carefully!) Add ct and add x to get the required form! 3E

  44. Coordinate Systems Start with the general equation of the hyperbola… You need to be able to find equations for tangents and normals algebraically The point P(ct,c/t), t ≠ 0, lies on the rectangular hyperbola H with equation xy = c2 where c is a positive constant. Show that an equation of the tangent to H at P is: • A rectangular hyperbola G has equation xy= 9. The tangent to G at A and the tangent to G at B meet at the point (-1,7). • Find the coordinates of A and B • The coordinates are in terms of c and t, so we will need to find the values of these… We have been told the actual equation We can therefore deduce the value of c2 Square root (we only need the positive value as we are told in the question that c is positive) 3E

  45. Coordinate Systems We can use the value for c, along with the coordinate given, to find the value of t We know an equation of the tangent, as well as a coordinate it passes through… You need to be able to find equations for tangents and normals algebraically The point P(ct,c/t), t ≠ 0, lies on the rectangular hyperbola H with equation xy = c2 where c is a positive constant. Show that an equation of the tangent to H at P is: • A rectangular hyperbola G has equation xy= 9. The tangent to G at A and the tangent to G at B meet at the point (-1,7). • Find the coordinates of A and B • The coordinates are in terms of c and t, so we will need to find the values of these… Sub in x, y and c Simplify terms Rearrange for solving 7 Factorise Find the values of t We have 2 values for t as there are 2 tangents that will pass through the coordinate (-1,7) 3E

  46. Coordinate Systems You need to be able to find equations for tangents and normals algebraically The point P(ct,c/t), t ≠ 0, lies on the rectangular hyperbola H with equation xy = c2 where c is a positive constant. Show that an equation of the tangent to H at P is: • A rectangular hyperbola G has equation xy= 9. The tangent to G at A and the tangent to G at B meet at the point (-1,7). • Find the coordinates of A and B • The coordinates are in terms of c and t, so we will need to find the values of these… Sub in c and t = -1/7 Calculate Sub in c and t = 1 Calculate 3E

  47. Summary • We have learnt what parametric equations are • We have learnt the general equation od a parabola, as well as seeing its focus and directrix • We have learnt the equation for a rectangular hyperbola • We have learnt to solve problems relating to the tangent and normal at points on a parabola or hyperbola • We have see how to approach algebraic versions of questions on the tangent and normal

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