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3 Further Transformations of the Ellipse, Hyperbola and Parabola

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3 Further Transformations of the Ellipse, Hyperbola and Parabola 3 Further transformations of the ellipse, hyperbola and parabola In this chapter you will learn how to: • recognise an ellipse, a hyperbola or a parabola from its parametric or polar equation • find the equation of a curve after a combination of rotations, reflections and stretches • identify a transformation, or a sequence of transformations, by considering the transformed equation. Before you start… Further Mathematics You should be able to recognise the 1 Name the curve described by the Student Book 1, equation and main features of an ellipse, x2 y2 equation − = 1 and find the Chapter 3 hyperbola and parabola. 16 9 equations of any asymptotes and the coordinates of any axis intercepts. Further Mathematics You should know how to find the 2 Find the equation of the curve Student Book 1, equation of a curve after a translation, y2 = 6x after: Chapter 3 reflection of stretch. −2 a a translation with vector ( 5 ) b a reflection in the line y = −x. A Level Mathematics You should be able to recognise a 3 Describe two transformations which Student Book 2, transformation, or combination of map the graph of y = f(x) onto the Chapter 3 transformations, that map one function graph of y = −f(3x). onto another. Conic sections Conic sections is a collective name for a group of curves which can be obtained as cross-sections of a cone. There are three distinct types: the ellipse (with a circle as a special case), hyperbola and parabola. In Further Mathematics Student Book 1, Chapter 3, you met Cartesian equations of theseDraft curves and also learnt how the equations sample are changed by translations, reflections and stretches. In this chapter you will add enlargements and rotations to this list of transformations. You will also see how you can apply techniques of parametric equations and polar coordinates to describe the curves in several different ways. © Cambridge University Press 2018 47 The third party copyright material that appears in this sample may still be pending clearance and may be subject to change. A Level Further Mathematics for AQA Student Book 2 Section 1: Parametric and polar form Ellipse Hyperbola Parabola y y y b b b y = – — x y = — x y2 = 4ax x2 y2 — + — = 1 a a x2 y2 a2 a2 — – — = 1 a2 a2 x x x –a O a –a O a O –b You can also describe these curves using parametric and polar equations. Rewind You met parametric equations in A Level Mathematics Student Book 2, Chapter 12, and polar coordinates in Further Mathematics Student Book 1, Chapter 6. WORKED EXAMPLE 3.1 Show that the curve described by the parametric equations 2 π π x = 1 + tan t, y = 3 sec t, t ∈ − , is a parabola. ( 2 2 ) 2 2 y = 9 sec t You need to find a Cartesian equation. It looks like = 9(1 + tan2 t) y2 is related to x. = 9x Hence the Cartesian equation is y2 = 9x, Recognise the standard equation of a parabola. which is the equation of a parabola. WORKED EXAMPLE 3.2 36 Find a Cartesian equation of the curve with polar equation r2 = . Hence sketch the curve. 13 cos2 q − 4 x Using cos q = r : The connection between polar and Cartesian 2 2 2 2 36 Draft samplecoordinates is: r = x + y , x = r cos q and r = 13x2 − 4 y = r sin q . It is often useful to start by expressing r2 cos q in terms of x and r and see if any r terms cancel. r2 ⇔ r2 = 36 13x2 − 4r2 Now use r2 x2 y2. ⇔ 13x2 − 4r2 = 36 = + ⇔ 13x2 − 4(x2 + y2) = 36 ⇔ 9x2 − 4y2 = 36 Divide by 36 to get 1 on the right. x2 y2 ⇔ − = 1 4 9 Continues on next page 48 © Cambridge University Press 2018 The third party copyright material that appears in this sample may still be pending clearance and may be subject to change. 3 Further transformations of the ellipse, hyperbola and parabola x2 y2 This is a hyperbola with x-intercepts (±2, 0) and This is the equation of a hyperbola − = 1 a2 b2 3 with a = 2 and b = 3. ( ) asymptotes y = ± x. 2 y 3 y = – — 3 x y = — x 2 2 x2 y2 — – — = 1 4 9 x (–2, 0) O (2, 0) Transformations of curves You already know how to apply certain transformations to curves by changing the equation. Transformation Change to equation p replace x by (x − p) and y translation with vector (q) by ( y − q) reflection in the x-axis replace y by −y reflection in the y-axis replace x by −x reflection in the line y = x swap x and y reflection in the line y = −x replace x by −y and y by −x stretch horizontally with scale factor y replace x by x and y by a and vertically with scale factor b ( a ) ( b ) WORKED EXAMPLE 3.3 3 Show that the curve with polar equation r = is an ellipse and state the equations of its axes of 2 + cos q symmetry. Using x = r cos q : Start by expressing cos q in terms of x and r. r r = 3 = 3 x 2r + x 2 + r ⇔ 2r + x = 3 Draft sampleNote that r can’t be zero, so you can divide by it. ⇔ 4r2 = (3 − x)2 ⇔ 4x2 + 4y2 = 9 − 6x + x2 r = √x2 + y2, so isolate r on the left before squaring ⇔ 3x2 + 6x + 4y2 = 9 both sides. (r is, by definition, positive, so squaring ⇔ 3(x + 1)2 − 3 + 4y2 = 9 both sides is fine.) ⇔ 3(x + 1)2 + 4y2 = 12 (x + 1)2 y2 You are looking for an equation of the ellipse, so ⇔ + = 1 4 3 complete the square for x and make the RHS equal to 1. Continues on next page © Cambridge University Press 2018 49 The third party copyright material that appears in this sample may still be pending clearance and may be subject to change. A Level Further Mathematics for AQA Student Book 2 −1 This is an ellipse translated −1 units horizontally. Recognise the translation with vector . y ( 0 ) (x + 1)2 y2 —+ — = 1 4 3 –1 y = 0 x –3 O 1 x = –1 Hence its axes of symmetry are y = 0 and x = −1. Did you know? The equation of an ellipse in the form shown in Worked Example 3.3 is normally used to describe planetary orbits. EXERCISE 3A 1 Show that the curve described by the parametric equations x = 3t2, y = 6t, t ∈ ℝ is a parabola. 2 Show that the curve with equations x = 2t2 + 1, y = 5t − 2 is a parabola and state the coordinates of its vertex. 3 Find the Cartesian equation of the curve with parametric equations x = 4 cos t, y = 5 sin t , t ∈ [0, 2π). Hence sketch the curve. 4 Show that the curve with parametric equations y = 7 cosh t, y = 3 sinh t is a hyperbola and find the equations of its asymptotes. 5 Sketch the curve with parametric equations x = 5 sec t, y = 5 tan t. 6 Show that the curve with parametric equations x = 3 − 5 cos q , y = 1 + 2sin q is an ellipse and state the equations of itsDraft axes of symmetry. sample 7 Show that the curve with parametric equations x = cos 2q, y = cos q , q ∈ [0, 2π) is a parabola. State the coordinates of its vertex. 5 8 Find the Cartesian equation of the curve with polar equation r2 = . What is the name given to this 2 cos2 q −1 curve? 2 9 Find the Cartesian equation of the polar curve r2 = . What is the name given to this curve? 3 sin2 q + 1 50 © Cambridge University Press 2018 The third party copyright material that appears in this sample may still be pending clearance and may be subject to change. 3 Further transformations of the ellipse, hyperbola and parabola 9 10 A curve is described by the polar equation r = . 1 + 2 cos q a Find the Cartesian equation of the curve. b Hence sketch the curve, indicating the position of the intercepts with the coordinate axes. 40 11 Find the Cartesian equation of the curve given by polar equation r = . State the name given to this 3 − sin q curve and write down the equations of its axes of symmetry. 2 2 12 Curve C1 has equation 5x + 12y = 32. a Find the equation of the tangent to C1 at the point (2, 1). Give your answer in the form ax + by = c, where a, b and c are integers. 2 2 Curve C2 has equation 5x − 30x + 12y + 48y + 61 = 0. b Describe the transformation which transforms C1 to C2. c Hence find the equation of the tangent to C2 at the point (5, −1). 13 An ellipse is obtained from the circle x2 + y2 = 18 by a horizontal stretch with scale factor 5. a Find the equation of the ellipse. b Write down the equation of the tangent to the circle at the point (3, 3). c Hence find the equation of the tangent to the ellipse at the point (15, 3). 2 14 The parabola C1 has equation y = 16(x − 2). Tip a Find the equations of the tangents to C1 which pass through the origin. There are several possible 2 answers to part b of Question The parabola C2 has parametric equations x = t + 8, y = 2t, t ∈ ℝ . 14. Try to find a transformation b Describe the transformations which transform C1 to C2. which is given exclusively in terms of stretches rather than c Hence find the equations of the tangents to C2 which pass through the origin.
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