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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

hyperbola

双曲线

hyperbola



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.

hyperbola


Simplest form of the eqn of a hyperbola

e:eccentricity

hyperbola


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

hyperbola


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 .

hyperbola


Soln:

(i)

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(ii)

Coordinates of the foci are

(iii) Equations of directrices are

(iv) Equations of asymptotes are

i.e.

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e.g. 2

Find the asymptotes of the hyperbola .

Soln:

The asymptotes are

hyperbola


e.g. 3

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

hyperbola


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.

hyperbola


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

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

hyperbola




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.

hyperbola


3.

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

4.

Asymptotes of the hyperbola :

hyperbola


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

hyperbola


1. corresponding results for the ellipse by merely writing in place of . 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

hyperbola


3. corresponding results for the ellipse by merely writing in place of . The locus of the midpoints of chords of the hyperbola with gradient m is the diameter :

hyperbola


e.g. 4 corresponding results for the ellipse by merely writing in place of .

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

hyperbola


Soln: corresponding results for the ellipse by merely writing in place of .

Gradient of tangents=2

Hence, equations of tangents are :

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

O

Distance=

hyperbola


The rectangular hyperbola corresponding results for the ellipse by merely writing in place of .

hyperbola


1. corresponding results for the ellipse by merely writing in place of . A hyperbola with perpendicular asymptotes is a rectangular hyperbola.

i.e. b=a

So, the standard equation of a rectangular hyperbola is :

hyperbola


2. corresponding results for the ellipse by merely writing in place of . Eccentricity of a rectangular hyperbola =

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y asymptotes

y

x

O

x

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Some simple sketches of the rectangular hyperbola : asymptotes

y

y

xy=-9

xy=9

x

o

x

o

y

y

x

x

o

o

hyperbola


hyperbola asymptotes



The equation, asymptotes

is satisfied if

t is a parameter.

The parametric coordinates of any point are :

hyperbola


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

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

hyperbola


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

hyperbola


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

hyperbola


e.g. 5 asymptotes

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.

hyperbola


Soln: asymptotes

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).

hyperbola



e.g. 6 asymptotes

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.

hyperbola


Soln: asymptotes

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

Gradient of tangent at P is :

S

R

P

at

hyperbola


Gradient of PS= asymptotes

Gradient of RP=

Gradient of RS=

Hence,

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Conclusion: asymptotes

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


In the asymptotesCartesian 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.

hyperbola


then: asymptotes

  • 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. )

hyperbola


Analyzing an Hyperbola asymptotes

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.

hyperbola


~ The end ~ asymptotes

hyperbola


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

Misc.14 no need to do

hyperbola


Ex 14d Q 1 asymptotes

At point (2a,a/2),

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

hyperbola


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

hyperbola


Ex 14d Q 3 asymptotes

(2t,2/t)

2y=x+7

4/t=2t+7

hyperbola


At point asymptotesA, 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

hyperbola


At point asymptotesA, dy/dx=-4

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

Eqn of tangent at A :

hyperbola


Eqn of tangent at asymptotesB :

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

128-64x+x+16=0

x=144/63=16/7

hyperbola


x=16/7 asymptotes

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

=8-64/7

=-8/7

Point of intersection is

(16/7, -8/7)

hyperbola


Ex 14d Q 5 asymptotes

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

4xy=25

hyperbola


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

3t -1

t 2

hyperbola


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

t=1/3 or t=-2

When t=1/3,

hyperbola


When asymptotest=-2,

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

hyperbola


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

m1=-1/4 and m2=-9

Hence,

hyperbola


Ex 14d Q 5 asymptotes

Why this method can’t be accepted???

(2,-3) 4xy=25

Equation of tangent lines in gradient form :

Therefore,

hyperbola


Let asymptotesm1 and m2 be the roots,

We have , and

hyperbola


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

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

We need to know m1-m2 now.

hyperbola


Sub. Into asymptotes1

hyperbola


Hence, the acute angle is . asymptotes

hyperbola


Ex 14d Q 7 asymptotes

xy=9 (-1,-9)

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

hyperbola


Hence, asymptotes

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

So, length of the chord is

hyperbola


Ex 14d Q 9 asymptotes

To find points of intersection of 2 hyperbola :

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

Sub. into 1

and

hyperbola


At asymptotes(1,4)

The product of these gradient =-1.

At (-1,-4)

Hence, the product of these gradient =-1.

hyperbola


y asymptotes

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

hyperbola


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

Hence, QP=PR

hyperbola


The 2009 year end examination scopes : asymptotes

  • The straight line

  • The circle

  • The parabola

  • The ellipse

  • The hyperbola 排列与组合

  • Permutations & combinations

  • Probability概率

hyperbola


2 asymptotesnd 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%

hyperbola


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