Further Coordinate Geometry

One of the best ways to learn how a concept works is to understand its proof.

Below are the proofs for coordinate geometry concepts co-ordinate geometry concepts.

Perpendicular Distance Formula

The line ax + by + c = 0 has gradient –a/b

ax₀ + by₁ + c = 0 (Point C lies on the line)

∴ x₀ = -(by₁ + c)/a

Distance of PC = (ax₁ + by₁ + c)/a

Sin θ = d/PC

d = PC sinθ

if tanθ = -a/b

sinθ = -a/√(a² + b²)

∴ d = [(ax₁ + by₁ + c)/a] x [-a/√(a² + b²)]

= -(ax₁ + by₁ + c)/√(a² + b²)

Since distance is positive, we take the absolute value of this so

|ax₁ + by₁ + c|/√(a² + b²)

Internal division of an interval

In ∆PAC and ∆BPD

∠PAC = ∠BPD (corresponding angles on parallel lines)

∠ APC = ∠ PBD (corresponding angles on parallel lines)

∴ ∆PAC ||| ∆BPD (equiangular)

(x-x₁)/(x₂-x) = k/l

l(x-x₁) = k(x₂-x)

kx +lx = lx₁ +kx₂

x = (lx₁ + kx₂)/k+l

Similarly

(y-y₁)/(y₂-y) = k/l

l(y-y₁) = k(y₂-y)

ky +ly = ly₁ +ky₂

y = (ly₁ + ky₂)/k+l

Angle between two intervals

Let the angle of inclination of l₁ be α₁

∴ tan α₁ = m₁

Let the angle of inclination of l₂ be α₂

∴ tanα₂ = m₂

Angle between the intervals = α₂ – α₁

tan (α₂ – α₁) = (tan α₂ – tan α₁)/1+tanα₁tanα₂

= (m₂ – m₁)/1+m₁m₂

Functions and relations

A function is a graph that has one y value for every x value. Hence a common test for whether a graph is a function is the vertical line test.

In function notation, instead of using y, we use f(x) which is more convenient for substituting values.

Eg. if f(x) = x²

f(2) = 2² = 4

f(x+h) = (x+h)²

Common graphs

Straight line graph

Parabola

Cubic graph

Hyperbola

Exponential

Logarithm

Square Root Graph

Semi Circle

Circle

Absolute Value Graph

Graphs can be translated, flipped and transformed.

The transformation of graphs is counter intuitive.

To move a graph along h units along the x axis we substitute f(x-h)

Eg. (x-2)² is an x² graph shifted along the x axis by 2 units on the positive side.

Similarly if we want to move the graph k units up the y axis, we subtract k from f(x)

Eg. f(x) – 2 = x² is an x² graph shifted up two units.

This is commonly written as x² + 2 but in this case, counter intuition no longer holds.

If we want to flip a graph along the x axis, we substitute –y

If we want to flip a graph along the y axis, we substitute –x

Even and odd functions

A function is:

Even if for every x value, f(x) = f(-x) that is that if the graph is flipped along the y axis, it would still be the same graph.

To test for even functions, we substitute –x into the original function and we check to see whether the resulting function is equal to the original.

Odd if for every x value, f(x) = -f(x) that is that the graph has rotational symmetry about the origin.

To test for odd functions, we substitute –x into the original and we check whether the resulting function is equal to –f(x).

When we substitute –x into f(x) and find it is no equal to f(x) or –f(x), then the function is neither even nor odd.