Chapter 33 hyperbola 11232020 hyperbola 1 11232020 hyperbola

Chapter 33 hyperbola 双曲线 11/23/2020 hyperbola 1

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Definition: The locus of a point P which moves such that the ratio of its distances from a fixed point S and from a fixed straight line ZQ is constant, e , and greater than one. S is the focus, ZQ the directrix and the e, eccentricity of the hyperbola. 11/23/2020 hyperbola 3

Simplest form of the eqn of a hyperbola e: eccentricity 11/23/2020 hyperbola 4

Q The foci S, S’ are the points (-ae, 0), (ae, 0). Q’ y The directrices ZQ, Z’Q’ are the lines x= A’ A x -a/e, x=a/e. O Z’ S’ S Z AA’ is called the 实轴 transverse axis=2 a. B’ B BB’ is called the 虚轴 conjugate axis=2 b. asymptotes 11/23/2020 hyperbola 5

e. g. 1 For the hyperbola , find (i) the eccentricity, (ii) the coordinates of the foci (iii) the equations of the directrices and (iv) the equations of the asymptotes. 11/23/2020 hyperbola 6

Soln: (i) hyperbola 7 11/23/2020

(ii) Coordinates of the foci are (iii) Equations of directrices are (iv) Equations of asymptotes are i. e. 11/23/2020 hyperbola 8

e. g. 2 Find the asymptotes of the hyperbola . Soln: The asymptotes are 11/23/2020 hyperbola 9

e. g. 3 Find the equation of hyperbola with focus (1, 1); directrix 2 x+2 y=1; e=. 11/23/2020 hyperbola 10

Soln: From definition of a hyperbola, we have PS=e. PM. Where PS is the distance from focus to a point P and PM is the distance from the directrix to a point P. e is the eccentricity of the hyperbola. 11/23/2020 hyperbola 11

Let P be (x, y). Hence, distance from P to (1, 1) is : Distance from P to 2 x+2 y-1=0 is : 11/23/2020 hyperbola 12

is the required equation of the hyperbola. 11/23/2020 hyperbola 13

14 Properties of the hyperbola 11/23/2020

1. The curve is symmetrical about both axes. The curve exists 2. for all values of y. The curve does not exist if |x|<a. 11/23/2020 hyperbola 15

3. At the point (a, 0) & (-a, 0), the gradients are infinite. 4. Asymptotes of the hyperbola : 11/23/2020 hyperbola 16

Many results for the hyperbola are obtained from the corresponding results for the ellipse by merely writing in place of . 11/23/2020 hyperbola 17

1. The equation of the tangent to the hyperbola at the point (x’, y’) is 2. The gradient form of the equation of the tangent to the hyperbola is 11/23/2020 hyperbola 18

3. The locus of the midpoints of chords of the hyperbola with gradient m is the diameter : 11/23/2020 hyperbola 19

e. g. 4 Show that there are two tangents to the hyperbola parallel to the line y=2 x-3 and find their distance apart. 11/23/2020 hyperbola 20

Soln: Gradient of tangents=2 Hence, equations of tangents are : O 11/23/2020 Perpendicular distance from (0, 0) to the lines are : Distance= hyperbola 21

The rectangular hyperbola 22 hyperbola 11/23/2020

1. A hyperbola with perpendicular asymptotes is a rectangular hyperbola. i. e. b=a So, the standard equation of a rectangular hyperbola is : 11/23/2020 hyperbola 23

2. Eccentricity of a rectangular hyperbola = 11/23/2020 hyperbola 24

Equation of a rectangular hyperbola with respect to its asymptotes 11/23/2020 hyperbola 25

y y x O x 11/23/2020 hyperbola 26

Some simple sketches of the rectangular hyperbola : y xy=-9 xy=9 x o y y y o 11/23/2020 x o x x o hyperbola 27

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Parametric equations of a rectangular hyperbola 11/23/2020 hyperbola 29

The equation, is satisfied if t is a parameter. The parametric coordinates of any point are : 11/23/2020 hyperbola 30

Tangent and normal at the point (ct, c/t) to the curve Gradient of tangent at (ct, c/t) is 11/23/2020 hyperbola 31

Equation of tangent at (ct, c/t) is 11/23/2020 hyperbola 32

Equation of normal at (ct, c/t) is 11/23/2020 hyperbola 33

e. g. 5 The tangent at any point P on the curve xy=4 meets the asymptotes at Q and R. Show that P is the midpoint of QR. 11/23/2020 hyperbola 34

Soln: Let P be the point (2 t, 2/t). Equation of tangent at P is x-axis and y-axis are the asymptotes. When y=0, Q is (4 t, 0), when x=0 R is (0, 4/t). The midpoint of QR is (2 t, 2/t). 11/23/2020 hyperbola 35

Miscellaneous examples on the hyperbola 11/23/2020 hyperbola 36

e. g. 6 A chord RS of the rectangular hyperbola subtends a right angle at a point P on the curve. Prove that RS is parallel to the normal at P. 11/23/2020 hyperbola 37

Soln: Let S(ct, c/t), R(cp, c/p) and P(cq, c/q). S Gradient of tangent at P is : R P at 11/23/2020 hyperbola 38

Gradient of PS= Gradient of RP= Gradient of RS= Hence, 11/23/2020 hyperbola 39

Conclusion: In analytic geometry, the hyperbola is represented by the implicit equation : The condition : B 2 − 4 AC > 0 (if A + C = 0, the equation represents a rectangular hyperbola. ) 11/23/2020 Ellipse 40

In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section, and all conic sections arise in this way. The equation will be of the form : 2 Ax + Bxy + + Dx + Ey + F = 0 with A, B, C not all zero. 11/23/2020 2 Cy hyperbola 41

then: • if B 2 − 4 AC < 0, the equation represents an ellipse (unless the conic is degenerate, for example x 2 + y 2 + 10 = 0); • if A = C and B = 0, the equation represents a circle; • if B 2 − 4 AC = 0, the equation represents a parabola; • if B 2 − 4 AC > 0, the equation represents a hyperbola; (if A + C = 0, the equation represents a rectangular hyperbola. ) 11/23/2020 hyperbola 42

Analyzing an Hyperbola State the coordinates of the vertices, the coordinates of the foci, the lengths of the transverse and conjugate axes and the equations of the asymptotes of the hyperbola defined by 4 x 2 - 9 y 2 + 32 x + 18 y + 91 = 0. 11/23/2020 hyperbola 43

~ The end ~ 11/23/2020 hyperbola 44

Ex 14 d do Q 1, 3, 5, 7, 9, 11. Misc. 14 no need to do 45 hyperbola 11/23/2020

Ex 14 d Q 1 At point (2 a, a/2), Gradient of normal at (2 a, a/2) is 4. 11/23/2020 hyperbola 46

Equation of normal at (2 a, a/2) is : 11/23/2020 hyperbola 47

Ex 14 d Q 3 (2 t, 2/t) 2 y=x+7 4/t=2 t+7 11/23/2020 hyperbola 48

At point A, t=1/2 so, A is (1, 4) At point B, t=-4 so, B is (-8, -1/2) x=2 t, y=2/t Hence, xy=4 11/23/2020 hyperbola 49

At point A, dy/dx=-4 At point B, dy/dx=-1/16 Eqn of tangent at A : 11/23/2020 hyperbola 50

Eqn of tangent at B : Hence, 16(8 -4 x)+x+16=0 128 -64 x+x+16=0 x=144/63=16/7 11/23/2020 hyperbola 51

x=16/7 Therefore, y=8 -4(16/7) =8 -64/7 =-8/7 Point of intersection is (16/7, -8/7) 11/23/2020 hyperbola 52

Ex 14 d Q 5 Let the point on the curve be (ct, c/t). 4 xy=25 11/23/2020 hyperbola 53

Eqn of tangent lines at (ct, c/t) : 3 t -1 t 2 11/23/2020 hyperbola 54

(3 t-1)(t+2)=0 t=1/3 or t=-2 When t=1/3, 11/23/2020 hyperbola 55

When t=-2, So, m 1=-1/4 and m 2=-9 11/23/2020 hyperbola 56

Let the two angles of these tangent lines be A and B. m 1=-1/4 and m 2=-9 Hence, 11/23/2020 hyperbola 57

Ex 14 d Q 5 (2, -3) Why this method can’t be accepted? ? ? 4 xy=25 Equation of tangent lines in gradient form : Therefore, 11/23/2020 hyperbola 58

Let m 1 and m 2 be the roots, We have , and 11/23/2020 hyperbola 59

Let the two angles between these tangent lines be A and B. -----------1 We need to know m 1 -m 2 now. 11/23/2020 hyperbola 60

Sub. Into 1 11/23/2020 hyperbola 61

Hence, the acute angle is . 11/23/2020 hyperbola 62

Ex 14 d Q 7 xy=9 (-1, -9) Eqn of normal line at (-1, -9) : 11/23/2020 hyperbola 63

Hence, Another point is y=1/9, x=81 So, length of the chord is 11/23/2020 hyperbola 64

Ex 14 d Q 9 To find points of intersection of 2 hyperbola : ------ 1 Sub. into 1 and 11/23/2020 hyperbola 65

At (1, 4) The product of these gradient =-1. At (-1, -4) Hence, the product of these gradient =-1. 11/23/2020 hyperbola 66

y Ex 14 d Q 11 Q y(x-1)=9 2 At P, y’=-9 Eqn of tangent line at P, At Q, x=1, y=18 At R, y=0, x=3 11/23/2020 P(2, 9) hyperbola R x (y-9)=(-9)(x-2) y=-9 x+27 67

P(2, 9), Q(1, 18), R(3, 0) Hence, QP=PR 11/23/2020 hyperbola 68

The 2009 year end examination scopes : 1. The straight line 2. The circle 3. The parabola 4. The ellipse 5. The hyperbola 排列与组合 6. Permutations & combinations 7. Probability概率 11/23/2020 hyperbola 69

nd 2 November, 2009 (Monday) S 2 S Mathematics Final Examination Part A : Short Questions -- answer all 9 questions x 5%=45% Part B : Long Questions -- answer 5 from 9 questions x 11%=55% 11/23/2020 hyperbola 70