Mathematics · Ch 5 — Straight Line

Distance between Two Parallel Lines

5.4.3

Distance between Two Parallel Lines

5.4.3 The Distance between Two Parallel Lines

Theorem. The distance between the parallel lines $ax+by+c_1=0$ and $ax+by+c_2=0$ (same $a,b$ , so genuinely parallel) is

p = \frac{|c_1-c_2|}{\sqrt{a^2+b^2}}.

Proof. To find the distance between two parallel lines, it is enough to take any one point on the first line and find its perpendicular distance from the second line (that distance is the same everywhere along a pair of parallel lines). The point $A\left(-\dfrac{c_1}{a},0\right)$ lies on the first line. Its distance from the second line $ax+by+c_2=0$ , by the point-to-line formula of 5.4.2, is

p = \frac{\left|a\left(-\frac{c_1}{a}\right)+b(0)+c_2\right|}{\sqrt{a^2+b^2}} = \frac{|c_2-c_1|}{\sqrt{a^2+b^2}} = \frac{|c_1-c_2|}{\sqrt{a^2+b^2}}.

$\blacksquare$ …

Figure 5.4.3-Fig5.17Fig. 5.17

What this figure shows. Diagram showing two parallel lines with the same a, b coefficients and different constants c1, c2, and the perpendicular distance p between them. This figure gives the reader a concrete visual reference for the geometric configuration described in the surrounding text, tying the abstract statement to …

Misc 5.4.3-Ex1Ex. — distance between two parallel lines

Worked out. Finds the distance between 6x + 8y + 21 = 0 and 3x + 4y + 7 = 0 by first scaling the second line to match the first's coefficients (6x+8y+14=0), then applying the parallel-lines distance formula p=|c1−c2|/√(a²+b²), obtaining 7/10. …

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