Introduction In this chapter you will learn how

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 C 4 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

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 2 9 4 1 0 1 4 9 y = 4 t -12 -8 -4 0 4 8 12 Now you have a set of coordinates, you can plot them on a standard Cartesian axis 3 A

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’) t -3 -2 -1 0 1 2 3 x = t 2 9 4 1 0 1 4 9 y = 4 t -12 -8 -4 0 4 8 12 10 y So: x = f(t) y = g(t) 10 Coordinates on a curve can be defined using parametric equations -10 x 3 A

Coordinate Systems You need to know what parametric equations are A curve has parametric equations: Divide by 2 a Now you can replace t in the other equation… 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… This is the general equation for a parabola! Replace t Square the fraction Combine terms ‘Cancel’ an ‘a’ term from the top and bottom Multiply by 4 a 3 A

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! Multiplying the equations together will actually cancel the ‘t’ terms straight away! Combine the right hand side ‘Cancel’ the ‘t’ terms Divide by x 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!) 3 A

Coordinate Systems You need to know the general equation of a parabola, as well as its focus-directrix properties y To the right is an example of a parabola with parametric equations: It will have a Cartesian equation in the form: (0, 0) x 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” 3 B

Parametric Form Cartesian Form Coordinate Systems You need to know the general equation of a parabola, as well as its focus-directrix properties Directrix y “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: Focus x The equation of the directrix is given by: or 3 B

Parametric Form Directrix Focus Cartesian Form Coordinate Systems 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 3 B

Parametric Form Directrix Focus Cartesian Form Coordinate Systems 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 3 B

Parametric Form Directrix Cartesian Form Coordinate Systems 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: Focus Directrix Substitute a = √ 2 Divide the coefficient of x by 4 to find the value of a = = = 3 B

Parametric Form Directrix Cartesian Form Coordinate Systems 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 y 2 = 4 ax, and find the value of a. P(x, y) D 6 x F (6, 0) x 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 x + 6 = 0 3 B

Parametric Form Directrix Focus Cartesian Form Coordinate Systems You need to know the general equation of a parabola, as well as its focus-directrix properties y D 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 y 2 = 4 ax, and find the value of a. P(x, y) y F 6 - x (6, 0) x 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! x + 6 = 0 3 B

Parametric Form Directrix Focus Cartesian Form Coordinate Systems y You need to know the general equation of a parabola, as well as its focus-directrix properties P(x, y) D 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 y 2 = 4 ax, and find the value of a. y F 6 - x (6, 0) x Square both sides Sub in values x + 6 = 0 Multiply out brackets Subtract 36, subtract x 2 Add 12 x The equation is y 2 = 24 x and the value of a = 6 3 B

Parametric Form Cartesian Form Directrix Coordinate Systems 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 y 2 = 8 x. The point S is the focus of the parabola. The straight line l passes through S and P. a) Find the coordinates of S Divide the coefficient of x by 4 to find ‘a’ Focus This allows us to write the coordinates of the focus Line l passes through (2, 0) and (8, -8) You can use one of the linear curve equations from C 1! b) Find an equation for l, giving your answer in the form ax + by + c = 0, where a, b and c are integers. Sub in the coordinates (carefully!) Simplify Cross-multiply Move terms so it is in the correct form Divide all by 2 to simplify 3 C

Parametric Form Cartesian Form Directrix Coordinate Systems The point P(8, -8) lies on the parabola C with equation y 2 = 8 x. The point S is the focus of the parabola. The straight line l passes through S and P. a) Find the coordinates of S b) Find an equation for l, giving your answer in the form ax + by + c = 0, where a, b and c are integers. 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 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 line l meets the parabola again at point Q Focus Double In terms of x Replace 8 x 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! 3 C

Parametric Form Cartesian Form Directrix Coordinate Systems 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 y 2 = 8 x. The point S is the focus of the parabola. The straight line l passes through S and P. a) Find the coordinates of S b) 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 Focus To find the midpoint of PQ you just find the mean of the x and y-coordinates… Sub in values Calculate Or you can use fractions! 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 3 C

Parametric Form Cartesian Form Directrix Coordinate Systems 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 y 2 = 8 x. The point S is the focus of the parabola. The straight line l passes through S and P. a) Find the coordinates of S b) Find an equation for l, giving your answer in the form ax + by + c = 0, where a, b and c are integers. l Q(0. 5, 2) Focus C y S(2, 0) x M(4. 25, -3) The line l meets the parabola again at point Q c) Find the coordinates of Q P(8, -8) 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 3 C

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 Double cone being sliced vertically 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 Asymptotes A rectangular hyperbola is one where the asymptotes are perpendicular to each other… 3 D

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: 3 D

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. Write in index form Differentiate Rewrite so it is easier to sub in values You will need to use differentiation from C 1/C 2 to find the gradient at P Divide by x At P, x = 2 Calculate 3 D

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. Sub in x = 2 Calculate You will need to use differentiation from C 1/C 2 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 3 D

Coordinate Systems You need to know an equation for a rectangular hyperbola and be able to find the equations of tangents and normals And use a formula from C 1… The point P, where x = 2, lies on the rectangular hyperbola H with equation xy = 8. Sub in the gradient and coordinate 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 C 1/C 2 to find the gradient at P Multiply out the bracket Add 2 x and subtract 4 to get the required form We also need a coordinate on the line to be able to work out its equation Work out the y-coordinate at P 3 D

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 C 1/C 2 to find the gradient at P Tangent The normal passes through the same coordinate, but the gradient will be different (as in C 1). Use -1/m Normal Work out the y-coordinate at P Sub in the gradient and coordinate We also need a coordinate on the line to be able to work out its equation Multiply out the bracket Double terms to remove the fraction Subtract 2 y and add 8 to get the required form 3 D

Coordinate Systems 2 x + 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. al m or 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. N t en ng Ta This is what we worked out! P(2, 4) x - 2 y + 6 = 0 Tangent Normal 3 D

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 y 2 = 27 x. The line l 1 is the tangent to C at A and the line l 2 is the tangent to C at B. Given that at A, y > 0: a) Find the coordinates of A and B b) Draw a sketch showing the parabola C, the points A and B and the lines l 1 and l 2 Sub in x = 3 Calculate Square root A must have the y-coordinate 9 as at A, y > 0 c) Find equations for l 1 and l 2, giving your answers in the form ax + by + c = 0 where a, b and c are integers 3 D

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 y 2 = 27 x. The line l 1 is the tangent to C at A and the line l 2 is the tangent to C at B. Given that at A, y > 0: a) Find the coordinates of A and B y A(3, 9) l 1 C x l 2 b) Draw a sketch showing the parabola C, the points A and B and the lines l 1 and l 2 c) Find equations for l 1 and l 2, giving your answers in the form ax + by + c = 0 where a, b and c are integers B(3, -9) 3 D

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 y 2 = 27 x. The line l 1 is the tangent to C at A and the line l 2 is the tangent to C at B. Given that at A, y > 0: a) Find the coordinates of A and B Square root Split the root up Split the 27 up Simplify and rewrite √x using indices b) Draw a sketch showing the parabola C, the points A and B and the lines l 1 and l 2 c) Find equations for l 1 and l 2, giving your answers in the form ax + by + c = 0 where a, b and c are integers To find an equation for l 1, we need the gradient at A (we already know the coordinate) Now we will need to differentiate to find the gradient function 3 D

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 y 2 = 27 x. The line l 1 is the tangent to C at A and the line l 2 is the tangent to C at B. Given that at A, y > 0: a) Find the coordinates of A and B Split terms b) Draw a sketch showing the parabola C, the points A and B and the lines l 1 and l 2 c) Find equations for l 1 and l 2, giving your answers in the form ax + by + c = 0 where a, b and c are integers To find an equation for l 1, we need the gradient at A (we already know the coordinate) Differentiate 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. 3 D

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 y 2 = 27 x. The line l 1 is the tangent to C at A and the line l 2 is the tangent to C at B. Given that at A, y > 0: Sub in x = 3 Cancel the √ 3 terms a) Find the coordinates of A and B y b) Draw a sketch showing the parabola C, the points A and B and the lines l 1 and l 2 c) Find equations for l 1 and l 2, giving your answers in the form ax + by + c = 0 where a, b and c are integers To find an equation for l 1, we need the gradient at A (we already know the coordinate) The tangent at A has a positive gradient so is 3/2 The tangent at B has a negative gradient so is -3/2 A(3, 9) l 1 C x l 2 B(3, -9) 3 D

Coordinate Systems Equation of line l 1 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 y 2 = 27 x. The line l 1 is the tangent to C at A and the line l 2 is the tangent to C at B. Given that at A, y > 0: a) Find the coordinates of A and B b) Draw a sketch showing the parabola C, the points A and B and the lines l 1 and l 2 c) Find equations for l 1 and l 2, giving your answers in the form ax + by + c = 0 where a, b and c are integers Sub in the gradient and coordinate Multiply out the bracket Multiply all terms by 2 Subtract 2 y and add 18 to get the required form 3 D

Coordinate Systems Equation of line l 2 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 y 2 = 27 x. The line l 1 is the tangent to C at A and the line l 2 is the tangent to C at B. Given that at A, y > 0: b) Draw a sketch showing the parabola C, the points A and B and the lines l 1 and l 2 c) Find equations for l 1 and l 2, giving your answers in the form ax + by + c = 0 where a, b and c are integers Sub in the gradient and coordinate a) Find the coordinates of A and B Multiply out the bracket Multiply all terms by 2 Add 3 x 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! 3 D

Coordinate Systems You need to be able to find equations for tangents and normals algebraically The point P(at 2, 2 at) lies on the parabola C with equation y 2 = 4 ax 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! 3 E

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

Coordinate Systems You need to be able to find equations for tangents and normals algebraically Sub these into the equation of a line formula (as you would do if they were numerical) The point P(at 2, 2 at) lies on the parabola C with equation y 2 = 4 ax 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 in the gradient and the coordinate Multiply out the bracket Add 2 at 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! 3 E

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 = c 2 where c is a positive constant. Show that an equation of the tangent to H at P is: Divide by x Write in a differentiatable form Differentiate – remember c 2 is just a number Rewrite x-2 This will be similar to the last question, only with a hyperbola instead of a parabola Combine Find an expression for the gradient at a point on the curve, by differentiating 3 E

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 = c 2 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 c 2 terms So the gradient of the tangent to H at P is given by: 3 E

Coordinate Systems You need to be able to find equations for tangents and normals algebraically Sub these into the equation of a line formula (as you would do if they were numerical) The point P(ct, c/t), t ≠ 0, lies on the rectangular hyperbola H with equation xy = c 2 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 coordinate and the gradient Multiply out the bracket Multiply all terms by t 2 (ensure you do this carefully!) Add ct and add x to get the required form! 3 E

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 = c 2 where c is a positive constant. Show that an equation of the tangent to H at P is: We have been told the actual equation We can therefore deduce the value of c 2 Square root (we only need the positive value as we are told in the question that c is positive) 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… 3 E

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 = c 2 where c is a positive constant. Sub in x, y and c Show that an equation of the tangent to H at P is: Simplify terms 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… Rearrange for solving 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) 3 E

Coordinate Systems You need to be able to find equations for tangents and normals algebraically P(ct, c/ The point t), t ≠ 0, lies on the rectangular hyperbola H with equation xy = c 2 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 3 E

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