ELLIPS

Why study ellipses?

Orbiting satellites (including the earth and the moon) trace out elliptical paths

One property of ellipses is that a sound (or any radiation) beginning in one focus of the ellipse will be reflected so it can be heard clearly at the other focus.

Ellipse

© Jill Britton, September 25, 2003

Statuary Hall in the U.S. Capital building is elliptic. It was in this room that John Quincy Adams, while a member of the House of Representatives, discovered this acoustical phenomenon. He situated his desk at a focal point of the elliptical ceiling, easily eavesdropping on the private conversations of other House members located near the other focal point.

Why are the foci of the ellipse important?

• St. Paul's Cathedral in London. If a person whispers near one focus, he can be heard at the other focus, although he cannot be heard at many places in between.

© 1994-2004 Kevin Matthews and Artifice, Inc. All Rights Reserved.

Untuk menggambar ellips dimulai dengan menggambar 2 titik fokus

Ellipses with Horizontal Major Axis

The ellipse is defined as the locus of a point (x,y) which moves so that the sum of its distances from two fixed points (called foci, or focuses ) is constant

Ellipses

An ellipse is the locus of a variable point on a plane so that the sum of its distance from two fixed points is a constant.

P’(x,y)

P’’(x,y)

Let PF1+PF2 = 2a where a > 0

aycxycx 2)()( 2222 2222 )(2)( ycxaycx

222222 )()(44)( ycxycxaaycx

222 44)(4 acxycxa 42222222 2)2( acxaxcycxcxa

42222222222 22 acxaxcyacaxcaxa

22422222 )( caayaxca

)()( 22222222 caayaxca 222 cabLet

222222 bayaxb

12

2

2

2

b

y

a

x standard equation of an ellipse

Ellipse with Vertical Major Axis

If the major axis is vertical, then the formula becomes:

We always choose our a and b such that a > b. The major axis is always associated with a.

Ellipses with Centre Other Than the OriginEllipses with Centre Other Than the Origin

Like the other conics, we can move the ellipse so the its axes are not on the x-axis and y-axis. We do this for convenience when solving certain problems. For the horizontal major axis case, if we move the intersection of the major and minor axes to the point (h, k), we have:

HIPERBOLAHIPERBOLA

Cooling towers for a nuclear power Cooling towers for a nuclear power plant have a hyperbolic plant have a hyperbolic

Throw 2 stones in a pond. The resulting concentric ripples meet in a hyperbola shape.

How are conics used in the real world?

• To build things such as a sculpture at Fermi National Accelerator Laboratory. You can see all the “standard conics”

Hyperbola

The huge chimney of a nuclear power plant has the shape of a hyperboloid, as does the architecture of the James S. McDonnell Planetarium of the St. Louis Science Center.

© Jill Britton, September 25, 2003

Hyperbola• The plane can intersect

two nappes of the cone resulting in a hyperbola.

Hyperbola - DefinitionA hyperbola is the set of all points in a plane such that the difference in the distances from two points (foci) is constant.

| d1 – d2 | is a constant value.

Hyperbola - DefinitionWhat is the constant value for the difference in the distance from the two foci? Let the two foci be (c, 0) and (-c, 0). The vertices are (a, 0) and (-a, 0).

| d1 – d2 | is the constant.

If the length of d2 is subtracted from the left side of d1, what is the length which remains?

| d1 – d2 | = 2a

Hyperbola - EquationFind the equation by setting the difference in the distance from the two foci equal to 2a.

| d1 – d2 | = 2a

2 2

22

1

2

( )

( )

x c y

d c

d

x y

Let |PF1-PF2| = 2a where a > 0

aycxycx 2|)()(| 2222 2222 )(2)( ycxaycx

222222 )()(44)( ycxycxaaycx

222 44)(4 acxycxa 42222222 2)2( acxaxcycxcxa

42222222222 22 acxaxcyacaxcaxa

42222222 )( acayaxac

)()( 22222222 acayaxac 222 acbLet

222222 bayaxb

12

2

2

2

b

y

a

x standard equation of a hyperbola

Parabola yang membuka ke timur-barat (east-west opening hyperbola)

Asymtot

a

b

Persamaan umum parabola yang membuka ke timur-barat

F(c,0)F(-c,0)

c2 = a2 + b2

Persamaan asymtotis

1

1

1

2

222

2

2

2

2

2

2

2

2

a

xby

a

x

b

y

b

y

a

x

22

222

22 )(

axa

by

axa

by

Untuk x besar nilai xax 22

Sehingga diperoleh persamaan asymtot xa

by

Parabola yang membuka ke utara selatanParabola yang membuka ke utara selatan

north-south opening hyperbolanorth-south opening hyperbola

Persamaan umum parabola yang membuka ke utara selatan

Persamaan asymtotnya

xb

ay

Our hyperbola may not be centred on (0, 0). In this case, we use the following formulas:

For a "north-south" opening hyperbola with centre (h, k), we have:

For an "east-west" opening hyperbola with centre (h, k), we have:

1. Find the coordinates of the vertices and foci of 25x2 + y2 = 25. Sketch the curve

Answer

So b = 1 and a = 5. In this example, the major axis is vertical.So the vertices are at (0, -5) and (0, 5). To find c, we proceed as before:

So the foci are at (0, -4.9) and (0, 4.9).

Sketch the hyperbola

Answer

First, we recognise that it is an north-south opening hyperbola, with a = 5 and b = 2 . It will look similar to Example 1 above, which was also a north-south opening hyperbola.We need to find:The y-intercepts (there are no x-intercepts for this example) Aymptotes: We have an east-west opening hyperbola, so the slopes of the asymptotes will be given by

xb

ay

In this example, a = 5 and b = 2. So the slopes of the asymptotes will be simply:-5/2 and 5/2. The equations for the asymptotes, since they pass through (0, 0), are given by:y = -5x/2 and y = 5x/2

Sketch the hyperbola

AnswerWe note that this is an "east-west opening" hyperbola, with a = 6 and b = 8.The center of this hyperbola will be at (2, -3), since h = 2 and k = -3 in this example.The best approach is to ignore the shifting (for now) and figure out the other parameters for the hyperbola.The vertices of the parabola are found when y = 0. This gives us x = -6 or x = 6.The asymptotes will have slope -8/6 = -4/3 OR 8/6 = 4/3.Now we can sketch the asymptotes and the vertices, remembering to shift everything so that the center is (2, -3):

The Earth revolves around the sun in an elliptical orbit, where the sun is at one of the foci. (This was discovered by Keppler in 1610). The major axis is approximately 149,600,000 km long and it is known that the ratio c/a is equal to 1/60. [The ratio c/a is called the eccentricity of the ellipse.](i) What are the greatest and least distances the Earth is from the sun?(ii) How far from the sun is the other focus?

(i) The closest we are to the sun is when we are at the vertex closest to the sun, and the furthest we are is when we are at the other vertex.Since 2a = 149,600,000, then a = 74,800,000.So the vertices are at (-74,800,000, 0) and (74,800,000, 0).We need to find c, to tell us where the foci are.

So the foci are at (-1,246,667, 0) and (1,246,667, 0).So the closest we are to the sun is-1,246,667 − (-74,800,000) = 73,553,333 kmThe furthest we are from the sun is:74,800,000 + 1,246,667 = 76,046,667 km.ii) The foci are 2 × 1,246,667 = 2,493,334 km apart. The radius of the sun is 1,400,000 km. Our orbit is almost circular.