Straight Lines & Pair of Lines

Linear Geometry Foundation

A linear equation in two variables $a x + b y + c = 0$ ($a^2 + b^2 \neq 0$) geometrically represents a straight line in the Cartesian plane $\mathbb{R}^2$.

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1. Distance Metrics & Angle Between Lines

Perpendicular Distance from $(x_1,y_1)$ to $ax+by+c=0$

$$d=\frac{|a{x}_1+b{y}_1+c|}{\sqrt{a^2+b^2}}$$

Distance Between Parallel Lines $ax+by+c_1=0$ and $ax+by+c_2=0$

$$d=\frac{|c_1-c_2|}{\sqrt{a^2+b^2}}$$

Angle $\theta$ Between Lines with Slopes $m_1,m_2$

$$\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$$

• Lines are parallel $⟺m_1=m_2$.
• Lines are perpendicular $⟺m_1{m}_2=-1$.

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2. Equations of Angle Bisectors

For lines $L_1:a_{1}x+b_{1}y+c_1=0$ and $L_2:a_{2}x+b_{2}y+c_2=0$ (make $c_1,c_2>0$):

$$\frac{a_{1}x+b_{1}y+c_1}{\sqrt{a_1^2+b_1^2}}=\pm \frac{a_{2}x+b_{2}y+c_2}{\sqrt{a_2^2+b_2^2}}$$

• Bisector Containing Origin: Choose $+$ sign.
• Acute vs Obtuse Angle Bisector:
  • If $a_1{a}_2+b_1{b}_2>0⟹+$ sign is Obtuse Bisector, $-$ sign is Acute Bisector.
  • If $a_1{a}_2+b_1{b}_2<0⟹+$ sign is Acute Bisector, $-$ sign is Obtuse Bisector.

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3. Homogenization of Second-Degree Curves

Homogenization Method

The joint equation of lines joining the origin to the points of intersection of a second-degree curve: $$a x^2 + 2h x y + b y^2 + 2g x + 2f y + c = 0$$ and a line $l x + m y + n = 0$ (written as $\frac{l x + m y}{-n} = 1$) is obtained by homogenizing to degree 2: $$a x^2 + 2h x y + b y^2 + 2(g x + f y)\left(\frac{l x + m y}{-n}\right) + c\left(\frac{l x + m y}{-n}\right)^2 = 0$$ These lines are mutually perpendicular if and only if: $$\text{Coefficient of } x^2 + \text{Coefficient of } y^2 = 0$$