MATH10001 Project 1

Conic Sections part 2

http://www.maths.manchester.ac.uk/undergraduate/ugstudies/units/2009-10/level1/MATH10001/

The ParabolaA parabola is the set of points in the plane that are equidistant from a point F and a line L.

F is called the focus and L is called the directrix.

The vertex of the parabola is the point on the parabola closest to F and L.

We start by looking at the parabola with vertex (0,0), focus (p,0) and directrix x = - p.

.4

gives gRearrangin

.)(

get we Since

2

22

pxy

ypxpx

|PF||PN|

N (-p,y) P(x,y)

F (p,0) x = -p

The HyperbolaA hyperbola with foci F1 and F2 is the set of points P in the plane such that

| |PF1| - |PF2| | = constant.

We start by assuming F1 = (-c,0) and F2 = (c,0).

.F2 (c,0)

.F1 (-c,0)

.P(x,y)

(a,0)

We have for some a.

This simplifies to

The eccentricity is e = c/a. In this case e > 1.

The directrices are x = a/e.

aycxycx 2)()( 2222

. where1 222

2

2

2 acb

b

y

a

x

(-a,0)

x = a/e x = -a/e

.F2 (c,0)

Proposition For any P on the right arm of the hyperbola

|PF2| = e.|PN|.

(A similar result holds for the left arm.)

N . . P(x,y)

Other hyperbolas

xy = 1

The assessment for this project is by an individual project report.

The report should contain your solutions to the four problems and the homework from part 1.

Even though you have worked as a group on this project, the report should be all your own work. There are marks for the clarity as well as the correctness of your mathematical arguments. Please hand in your report to your postgraduatefacilitator at the start of the group work class on Tuesday 20th October. You should attach a School cover sheet to your report.

There are 40 marks for this project.(a) 30 marks for the solutions to the problems.(b) 5 marks for the homework.(c) 1/6 of the average mark for your group for (a) out of 5.

Any student who does not attend one group session, without good reason, will get half the marks for (c). If a student misses both group sessions, without good reason, they will get 0 marks for (c). If you miss a session because of illness or personal problems pleasefill in a Self Certification Form available from the Teaching and Learning office reception in Alan Turing Building.