ANALYTIC GEOMETRY TIME A SMALL INTRODUCTION IN SOME REALLY INTERESTING TOPICS

1.BASIC DEFINITIONS Distance Formula Recall Pythagoras’ Theorem For a right-angled triangle with hypotenuse length c, We use this to find the distance between any two points (x1, y1) and (x2, y2) on the cartesian plane.

THE CARTESIAN PLANE

GRADIENT (OR SLOPE) The gradient of a line is defined as: In this triangle, the gradient of the line is given by:

In general, for the line joining the points (x1, y1) and (x2, y2):

POSITIVE AND NEGATIVE SLOPES In general, a positive slope indicates the value of the dependent variable increases as we go left to right: The dependent variable in the above graph is the y-value.

A negative slope means that the value of the dependent variable is decreasing as we go left to right:

INCLINATION We have a line with slope m and the angle that the line makes with the x-axis is α. From trigonometry, we recall that the tan of angle α is given by:

Now, since slope is also defined as opposite/adjacent, we have: This gives as the result: tan α = m

PARALLEL LINES Lines which have the same slope are parallel. If a line has slope m 1 and another line has slope m 2 then the lines are parallel if : m 1 = m 2

PERPENDICULAR LINES If a line has slope m1 and another line has slope m2 then the lines are perpendicular if: m1 × m2= -1

2.THE STRAIGHT LINE Slope-Intercept Form of a Straight Line The slope-intercept form (otherwise known as "gradient, y intercept" form) of a line is given by: y = mx + b

Point-Slope Form of a Straight Line We need other forms of the straight line as well. A useful form is the point-slope form (or point - gradient form). We use this form when we need to find the equation of a line passing through a point (x1, y1) with slope m: y − y 1 = m(x − x 1 )

GENERAL FORM Another form of the straight line which we come across is general form: Ax + By + C = 0

3.THE CIRCLE The circle with centre (0, 0) and radius r has the equation: x 2 + y 2 = r 2

(x − h) 2 + (y − k) 2 = r 2

The General Form Of The Circle. An equation which can be written in the following form (with constants D, E, F) represents a circle: x 2 + y 2 + Dx + Ey + F = 0

4.THE PARABOLA

Why do we study the parabola? The parabola has many applications in situations where: Radiation needs to be concentrated at one point (e.g. radio telescopes, pay TV dishes, solar radiation collectors) or Radiation needs to be transmitted from a single point into a wide parallel beam (e.g. headlight reflectors).

Definition of a Parabola: The parabola is defined as the locus of a point which moves so that it is always the same distance from a fixed point (called the focus) and a given line (called the directrix).

The Formula for a Parabola-Vertical Axis

Parabolas with Horizontal Axis y 2 = 4px

5.THE ELLIPSE

Why do we study ellipses? Orbiting satellites (including the earth and the moon) trace out elliptical paths.

Ellipses with Horizontal Major Axis The equation for an ellipse with a horizontal major axis is given by:

Ellipse with Vertical Major Axis: Our first example above had a horizontal 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.

6.THE HYPERBOLA How do we use the hyperbola? Navigation: Ship's navigators can plot their position by comparing GPS signals from different satellites. The technique involves hyperbolas. Physics: The movement of objects in space and of subatomic particles trace out hyperbolas in certain situations. Sundials: Historically, sundials made use of hyperbolas. Place a stick in the ground and trace out the path made by the shadow of the tip, and you'll get a hyperbola. Construction: Nuclear power plant smoke stacks have a hyperbolic cross section as illustrated above. Such 3-dimensional objects are called hyperboloids.

An asymptote is a line that forms a "barrier" to a curve. The curve gets closer and closer to an asymptote, but does not touch it.

Technical Definition of a Hyperbola: An hyperbola is the locus of points where the difference in the distance to two fixed foci is constant