Chapter 33
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Chapter 33. hyperbola. 双曲线. 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 .

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Chapter 33

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Chapter 33

Chapter 33

hyperbola

双曲线

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Chapter 33

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Chapter 33

Definition:

The locusof a point P which moves such that the ratio of its distances from a fixed point Sand 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.

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Chapter 33

Simplest form of the eqn of a hyperbola

e:eccentricity

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Chapter 33

The foci S, S’ are the points (-ae,0), (ae,0) .

Q

Q’

y

The directrices ZQ, Z’Q’ are the lines x=-a/e, x=a/e .

B

A’

A

x

O

Z’

S

Z

S’

AA’ is called the transverse axis=2a .

实轴

B’

BB’ is called the conjugate axis=2b .

虚轴

asymptotes

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Chapter 33

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 .

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Chapter 33

Soln:

(i)

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Chapter 33

(ii)

Coordinates of the foci are

(iii) Equations of directrices are

(iv) Equations of asymptotes are

i.e.

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Chapter 33

e.g. 2

Find the asymptotes of the hyperbola .

Soln:

The asymptotes are

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Chapter 33

e.g. 3

Find the equation of hyperbola with focus (1,1); directrix 2x+2y=1; e= .

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Chapter 33

Soln:

From definition of a hyperbola, we have PS=ePM .

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.

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Chapter 33

Let P be (x,y). Hence, distance from P to (1,1) is :

Distance from P to 2x+2y-1=0 is :

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Chapter 33

is the required equation of the hyperbola.

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Chapter 33

Properties of the hyperbola

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Chapter 33

1. The curve is symmetrical about both axes.

The curve exists for all values of y.

2.

The curve does not exist if |x|<a.

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Chapter 33

3.

At the point (a,0) & (-a,0), the gradients are infinite.

4.

Asymptotes of the hyperbola :

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Chapter 33

Many results for the hyperbola are obtained from the corresponding results for the ellipse by merely writing in place of .

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Chapter 33

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

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Chapter 33

3. The locus of the midpoints of chords of the hyperbola with gradient m is the diameter :

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Chapter 33

e.g. 4

Show that there are two tangents to the hyperbola parallel to the line y=2x-3 and find their distance apart.

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Chapter 33

Soln:

Gradient of tangents=2

Hence, equations of tangents are :

Perpendicular distance from (0,0) to the lines are :

O

Distance=

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Chapter 33

The rectangular hyperbola

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Chapter 33

1. A hyperbola with perpendicular asymptotes is a rectangular hyperbola.

i.e. b=a

So, the standard equation of a rectangular hyperbola is :

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Chapter 33

2. Eccentricity of a rectangular hyperbola =

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Chapter 33

Equation of a rectangular hyperbola with respect to its asymptotes

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Chapter 33

y

y

x

O

x

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Chapter 33

Some simple sketches of the rectangular hyperbola :

y

y

xy=-9

xy=9

x

o

x

o

y

y

x

x

o

o

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Chapter 33

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Chapter 33

Parametric equations of a rectangular hyperbola

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Chapter 33

The equation,

is satisfied if

t is a parameter.

The parametric coordinates of any point are :

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Chapter 33

Tangent and normal at the point (ct,c/t) to the curve

Gradient of tangent at (ct,c/t) is

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Chapter 33

Equation of tangent at (ct,c/t) is

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Chapter 33

Equation of normal at (ct,c/t) is

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Chapter 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.

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Chapter 33

Soln:

Let P be the point (2t,2/t).

Equation of tangent at P is

x-axis and y-axis are the asymptotes.

When y=0,Q is (4t,0), when x=0 R is (0,4/t).

The midpoint of QR is (2t,2/t).

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Chapter 33

Miscellaneous examples on the hyperbola

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Chapter 33

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.

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Chapter 33

Soln:

Let S(ct,c/t), R(cp,c/p) and P(cq,c/q).

Gradient of tangent at P is :

S

R

P

at

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Chapter 33

Gradient of PS=

Gradient of RP=

Gradient of RS=

Hence,

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Chapter 33

Conclusion:

In analytic geometry, the hyperbola is represented by the implicit equation :

The condition : B2 − 4AC > 0

  • (if A + C = 0, the equation represents a rectangular hyperbola. )

Ellipse


Chapter 33

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 :

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0with A, B, Cnot all zero.

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Chapter 33

then:

  • if B2 − 4AC < 0, the equation represents an ellipse (unless the conic is degenerate, for example x2 + y2 + 10 = 0);

  • ifA = C and B = 0, the equation represents a circle;

  • if B2 − 4AC = 0, the equation represents a parabola;

  • if B2 − 4AC > 0, the equation represents a hyperbola;

    • (if A + C = 0, the equation represents a rectangular hyperbola. )

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Chapter 33

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 4x2 - 9y2 + 32x + 18y + 91 = 0.

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Chapter 33

~ The end ~

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Chapter 33

Ex 14d do Q1, 3, 5, 7, 9, 11.

Misc.14 no need to do

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Chapter 33

Ex 14d Q 1

At point (2a,a/2),

Gradient of normal at (2a,a/2) is 4.

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Chapter 33

Equation of normal at (2a,a/2) is :

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Chapter 33

Ex 14d Q 3

(2t,2/t)

2y=x+7

4/t=2t+7

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Chapter 33

At point A, t=1/2 so, A is (1,4)

At point B, t=-4 so, B is (-8,-1/2)

x=2t, y=2/t

Hence, xy=4

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Chapter 33

At point A, dy/dx=-4

At point B,dy/dx=-1/16

Eqn of tangent at A :

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Chapter 33

Eqn of tangent at B :

Hence, 16(8-4x)+x+16=0

128-64x+x+16=0

x=144/63=16/7

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Chapter 33

x=16/7

Therefore, y=8-4(16/7)

=8-64/7

=-8/7

Point of intersection is

(16/7, -8/7)

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Chapter 33

Ex 14d Q 5

Let the point on the curve be (ct,c/t) .

4xy=25

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Chapter 33

Eqn of tangent lines at (ct,c/t) :

3t -1

t 2

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Chapter 33

(3t-1)(t+2)=0

t=1/3 or t=-2

When t=1/3,

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Chapter 33

When t=-2,

So, m1=-1/4 and m2=-9

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Chapter 33

Let the two angles of these tangent lines be A and B.

m1=-1/4 and m2=-9

Hence,

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Chapter 33

Ex 14d Q 5

Why this method can’t be accepted???

(2,-3) 4xy=25

Equation of tangent lines in gradient form :

Therefore,

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Chapter 33

Let m1 and m2 be the roots,

We have , and

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Chapter 33

Let the two angles between these tangent lines be A and B.

---------------------1

We need to know m1-m2 now.

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Chapter 33

Sub. Into 1

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Chapter 33

Hence, the acute angle is .

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Chapter 33

Ex 14d Q 7

xy=9 (-1,-9)

Eqn of normal line at (-1,-9) :

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Chapter 33

Hence,

Another point is y=1/9, x=81

So, length of the chord is

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Chapter 33

Ex 14d Q 9

To find points of intersection of 2 hyperbola :

----------- 1

Sub. into 1

and

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Chapter 33

At (1,4)

The product of these gradient =-1.

At (-1,-4)

Hence, the product of these gradient =-1.

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Chapter 33

y

Ex 14d Q 11

Q

P(2,9)

y(x-1)=9

x

R

2

At P, y’=-9

Eqn of tangent line at P, (y-9)=(-9)(x-2)

y=-9x+27

At Q, x=1, y=18

At R, y=0, x=3

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Chapter 33

P(2,9), Q(1,18), R(3,0)

Hence, QP=PR

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Chapter 33

The 2009 year end examination scopes :

  • The straight line

  • The circle

  • The parabola

  • The ellipse

  • The hyperbola 排列与组合

  • Permutations & combinations

  • Probability概率

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Chapter 33

2nd November, 2009 (Monday) S2S Mathematics Final Examination

Part A : Short Questions -- answer all 9 questions x 5%=45%

Part B : Long Questions -- answer 5 from 9questions x 11%=55%

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