Co-Ordinate Geometry: The Line

• Distance between two points: |PQ| = √(x₂-x₁)² + (y₂-y₁)²

• Midpoint of a line segment: (x₁+x₂/2, y₁+y₂/2)

Slope of a line

• With two points (x,y₁) and (x₂,y₂) slope = m = y₂-y₁/x₂-x₁
• In form y=mx+c slope = m
• In form ax + by + c = 0 slope = -a/b

Equation of a line formula: y - y₁ = m (x-x₁)

• a∥b ⟺ m₁=m₂ (parallel lines equal slopes)
• a⊥b ⟺ m₁ m₂ = -1

Derivation of the Divisor of Line Segment

• (x₁,y₁) (x₂,y₂) and R(x,y) are three points
• R divides line segment created by other two points internally in the ratio a

R = (bx₁+ax₂/b+a, by₁+ay₂/b+a)

• Note: If ratio was 1:1 we are talking about midpoint

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Co-Ordinate Geometry: The Line

Equation of line

• Formula above or y=mx+c (m = slope, c = y - intercept)
• Parallel lines have equal slopes so x and y coefficients are in same ratio
• Line parallel to ax+by+c = 0 is ax + by + d = 0 (where d ∈R)
• Perpendicular lines slopes multiply to give -1
• Line perpendicular to ax + by + c = 0 is bx - ay + d = 0 (where d ∈R)

Area of Triangle

• To use following formula one vertex must be (0,0)
• Let other vertex be (x₁, y₁), (x₂, y₂)

$\frac{1}{2} |x_1y_2 - x_2y_1|$

• If no vertex at (0,0) - translate one point to (0,0) and apply translation to the other two points.

Perpendicular Distance from a Point to a Line

• Perpendicular distance from point to line is the shortest distance.
• Distance from point (x₁, y₁) to line ax + by + c = 0 :

$d = \frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}$

• Equation of line must be in form ax + by + c = 0

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Co-Ordinate Geometry: The Line

Angle of inclination and angle between two lines

• Angle of inclination - anti - clockwise angle formed between a line and the positive sense of the x-axis
• Slope and angle m = tan α
• Horizontal lines - slope = 0, angle of inclination 0°
• Vertical lines have angle of inclination of 90°, slope is not defined if two lines p and q have slopes $m_1$ and $m_2$ respectively, and θ is the angle between them, then :

$\tan \theta = \pm \frac{m_1-m_2}{1+m_1m_2}$

$\tan \theta = \left|\frac{m_1-m_2}{1+m_1m_2}\right| \text{ acute angle}$

Can also be used to find the slope of a line if is known and one slope is known

Find $m_1$ using formula.