Mathematics · Ch 5 — Straight Line

Two-Points Form

5.3.3

Two-Points Form

5.3.3 Two-Points Form

Goal. Find the equation of the line through two known points $A(x_1,y_1)$ and $B(x_2,y_2)$ .

Proof. Let $P(x,y)$ be any third point on the line. Since $A,P$ both lie on the line, its slope equals $\dfrac{y-y_1}{x-x_1}$ ; since $A,B$ both lie on the line, its slope also equals $\dfrac{y_2-y_1}{x_2-x_1}$ . Equating,

\frac{y-y_1}{x-x_1} = \frac{y_2-y_1}{x_2-x_1},

which can be rearranged to the symmetric form

\frac{x-x_1}{x_1-x_2} = \frac{y-y_1}{y_1-y_2}.

This requires the line to be genuinely non-vertical and non-horizontal, i.e. $x_1 \ne x_2$ and $y_1 \ne y_2$ — the vertical/horizontal cases are handled separately (Section 5.3, $x=k$ or $y=k$ ). …

Figure 5.3.3-Fig5.11Fig. 5.11

What this figure shows. Diagram showing the two fixed points A(x1,y1), B(x2,y2) and a general point P(x,y) on the line joining them, used to derive the two-points form. This figure gives the reader a concrete visual reference for the geometric configuration described in the surrounding text, tying the abstract statement to …

Misc 5.3.3-Ex1Worked example — two-points form

Worked out. Finds the equation of the line through A(2,1) and B(1,2) by substituting directly into the two-points formula and simplifying to x + y − 3 = 0. Working through this worked example after reading the theory above helps consolidate the method before attempting the exercise questions that follow it in the …

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