- 68 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' Chapter 33' - garson

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

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

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

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

Coordinates of the foci are

(iii) Equations of directrices are

(iv) Equations of asymptotes are

i.e.

hyperbola

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

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

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

hyperbola

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

hyperbola

Gradient of tangents=2

Hence, equations of tangents are :

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

O

Distance=

hyperbola

hyperbola

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

i.e. b=a

So, the standard equation of a rectangular hyperbola is :

hyperbola

Equation of a rectangular hyperbola with respect to its asymptotes

hyperbola

is satisfied if

t is a parameter.

The parametric coordinates of any point are :

hyperbola

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

hyperbola

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

hyperbola

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

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

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

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 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.

hyperbola

- 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

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

hyperbola

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

hyperbola

Why this method can’t be accepted???

(2,-3) 4xy=25

Equation of tangent lines in gradient form :

Therefore,

hyperbola

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

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

We need to know m1-m2 now.

hyperbola

hyperbola

hyperbola

The product of these gradient =-1.

At (-1,-4)

Hence, the product of these gradient =-1.

hyperbola

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

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

hyperbola

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%

hyperbola

Download Presentation

Connecting to Server..