Relative Positions of Two Lines

Idea

Two lines in a plane can only do three things. They cross at one point, they never meet, or they are the same line.

You can tell which by counting the points they share: one, none, or all of them. There is no fourth case, so one test settles it.

Rule

For $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ , compare the ratios of coefficients:

\frac{a_1}{a_2} \ne \frac{b_1}{b_2} \Rightarrow \text{they intersect at one point}

\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2} \Rightarrow \text{parallel, never meeting}

\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \Rightarrow \text{the same line twice}

How it is used

The ratio test and the slope test give the same answer. Use whichever suits the form you are given.

Condition	Slope view	Shared points
$\dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2}$	different slopes	exactly one
$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \ne \dfrac{c_1}{c_2}$	equal slopes, different intercepts	none
$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$	equal slopes and equal intercepts	every point

Always take the ratios in the same order: first coefficients, then second coefficients, then constants. Keep the same line on top every time.

The most common mistake is comparing $\dfrac{a_1}{a_2}$ with $\dfrac{b_2}{b_1}$ , with the lines swapped.

This also explains what happens when you solve the two equations.

Different slopes give one solution. Equal slopes with different constants give something false, such as $0 = 7$ , which means no shared points. If all three ratios are equal you get $0 = 0$ , which means every point is shared.

Worked example

Decide how the lines $2x - 3y + 1 = 0$ and $4x - 6y + 5 = 0$ are positioned. Compare the first coefficients: $\dfrac{2}{4} = \dfrac{1}{2}$ . Compare the second: $\dfrac{-3}{-6} = \dfrac{1}{2}$ , which matches, so the slopes are equal. Now the constants: $\dfrac{1}{5}$ , which does not match $\dfrac{1}{2}$ . Equal in the first two ratios but not the third means parallel and distinct:

\frac{2}{4} = \frac{-3}{-6} \ne \frac{1}{5}

Explore

Drag C and D until the verdict says Parallel. Now move one of them by a single step. The verdict changes to Intersecting and a meeting point appears. Getting Coincident is harder: both C and D must sit exactly on the blue line.

Two lines, one plane

Try this. Use A(0,0), B(1,1), C(0,1), D(1,2) for parallel lines. Change D to (1,0) for perpendicular lines.

Point A · x-2Point A · y3Point B · x3Point B · y1Point C · x-3Point C · y-2Point D · x3Point D · y4

Blue slope: -0.4 · Green slope: 1 · Intersecting at (0.857, 1.857).