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