Equations of a Line and Distances

What pins a line down?#

You need sur­pris­ingly lit­tle to fix a line: a point and a slope will do, or two points, or the places where it cuts the axes. Each of these gives its own form of the equa­tion, and we will write them all down. Then you will learn to turn the gen­eral equa­tion $Ax + By + C = 0$ into whichever form is handy, and to mea­sure how far a point is from a line, and how far apart two par­al­lel lines are.

The stan­dard forms#

• Hor­i­zon­tal / ver­ti­cal: $y = b$; $x = a$.
• Point–slope: if the line passes through $(x_0, y_0)$ with slope $m$, then $y - y_0 = m(x - x_0)$.
• Two-point: $\displaystyle y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$.
• Slope–inter­cept: $y = mx + c$ ($c$ the $y$-inter­cept); or $y = m(x - d)$ with $x$-inter­cept $d$.
• Inter­cept form: $\displaystyle \frac{x}{a} + \frac{y}{b} = 1$, for a line cut­ting the axes at $(a, 0)$ and $(0, b)$.
• Gen­eral form: $Ax + By + C = 0$ ($A, B$ not both $0$): slope $\displaystyle -\tfrac{A}{B}$, $x$-inter­cept $\displaystyle -\tfrac{C}{A}$, $y$-inter­cept $\displaystyle -\tfrac{C}{B}$.

Dis­tance from a point to a line#

The per­pen­dic­u­lar dis­tance of $(x_1, y_1)$ from $Ax + By + C = 0$ is

$$\displaystyle d = \frac{\lvert Ax_1 + By_1 + C \rvert}{\sqrt{A^2 + B^2}}.$$

For two par­al­lel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$:

$$\displaystyle d = \frac{\lvert C_1 - C_2 \rvert}{\sqrt{A^2 + B^2}}.$$

Try these your­self#

1. Find the equa­tion of each line: (a) through $(0, -2)$ with slope $\displaystyle \tfrac{3}{4}$; (b) through $(-3, 5)$ and $(1, -3)$; (c) inter­cepts $-2$ and $5$; (d) through $(4, -3)$ par­al­lel to the $y$-axis.
2. Write $5x - 2y + 10 = 0$ in slope–inter­cept and inter­cept forms.
3. Find the line through $(1, -2)$ per­pen­dic­u­lar to $2x - 3y + 7 = 0$.
4. Find the line through $(2, 3)$ par­al­lel to $x + 4y = 9$.
5. Find the dis­tance of $(3, -5)$ from $4x - 3y + 8 = 0$.
6. Find the dis­tance between $x + 2y = 3$ and $2x + 4y = 11$.
7. Find the equa­tion of the line whose seg­ment between the axes is bisected at $(2, 3)$.
8. Find $k$ if the lines $kx + 2y = 5$ and $3x - 6y = 1$ are per­pen­dic­u­lar.
9. Find the equa­tion of the alti­tude from $A$ in tri­an­gle $A(2, 4)$, $B(-1, 1)$, $C(5, -2)$.
10. The Fahren­heit tem­per­a­ture is a lin­ear func­tion of the Cel­sius tem­per­a­ture, with $F = 32$ at $C = 0$ and $F = 212$ at $C = 100$. Write the equa­tion and find $C$ when $F = 104$.

Answers to check against#