Basic Definitions

If two points $A(x_1,y_1)$ and $B(x_2,y_2)$ are given, the point $C(x,y)$ that divides $\overline{AB}$ so that $AC:CB=\lambda:1$ (directed segments) has coordinates

Here $\lambda>0$ gives a point of the segment (internal division; $\lambda=1$ is the midpoint), and $\lambda<0,\ \lambda\neq-1$ gives a point on the line outside the segment (external division).

Every linear equation $Ax+By+C=0$, with $A$ and $B$ not both zero, represents a line in the coordinate plane. This is called the general equation of a line. Particular cases:

• $C=0,\ A\neq0,\ B\neq0$: the line $Ax+By=0$ passes through the origin.
• $A=0,\ B\neq0,\ C\neq0$: the line $By+C=0$ is parallel to the $Ox$ axis (the $x$-axis).
• $B=0,\ A\neq0,\ C\neq0$: the line $Ax+C=0$ is parallel to the $Oy$ axis (the $y$-axis).
• $B=C=0,\ A\neq0$: the line $Ax=0$ coincides with the $Oy$ axis.
• $A=C=0,\ B\neq0$: the line $By=0$ coincides with the $Ox$ axis.

When $B\neq0$ (the line is not vertical) the equation can be written $y=kx+b$, where $b=-\dfrac{C}{B}$ and $k=-\dfrac{A}{B}=\tan\alpha$. Here $A,\,B,\,C$ are the coefficients of the general equation of the line, and $\alpha$ is the angle formed by the line with the positive $x$-axis. The constant $b$ is the $y$-coordinate of the $y$-intercept. A vertical line $x=x_1$ ($B=0$) has no slope-intercept form.

For two distinct points $M_1(x_1,y_1)$, $M_2(x_2,y_2)$ the line is

i.e. $(x-x_1)(y_2-y_1)-(y-y_1)(x_2-x_1)=0$. When $x_1\neq x_2$ and $y_1\neq y_2$ this is the same as

If $x_1=x_2$ the line is vertical, $x=x_1$; if $y_1=y_2$ it is horizontal, $y=y_1$ — both are covered by the determinant form above.

An equation of the form

where $a=-\dfrac{C}{A}$ and $b=-\dfrac{C}{B}$ (with $A,\,B,\,C$ the coefficients of the general equation of the line), is called the equation of a line in intercept form. Here $a$ is the $x$-intercept — where the line meets the $x$-axis — and $b$ is the $y$-intercept — where the line meets the $y$-axis. This form requires $A,\,B,\,C$ all nonzero; a line through the origin or parallel to an axis has no such form.

An equation of the form

is called the normal equation of a line. Here $p$ is the length of the perpendicular dropped from the origin onto the line, and $\varphi$ is the angle formed by this perpendicular with the positive direction of the $Ox$ axis.

To obtain the normal equation from the general equation $Ax+By+C=0$, both sides are multiplied by the normalising factor

The sign before the radical is chosen so that $\mu C\le 0$. When $C=0$ (a line through the origin) $p=0$ and either sign gives a valid normal equation.

The angle $\varphi$ between the lines $y=k_1x+b_1$ and $y=k_2x+b_2$ $(0^\circ\le\varphi\le90^\circ)$ is found by the formula

The lines are parallel when $k_2=k_1$, and perpendicular when $k_1k_2=-1$.

If the lines are given by the equations $A_1x+B_1y+C_1=0$ and $A_2x+B_2y+C_2=0$, the angle $\varphi$ $(0^\circ\le\varphi\le90^\circ)$ between them satisfies

The lines are parallel when $A_1B_2-A_2B_1=0$, and perpendicular when $A_1A_2+B_1B_2=0$ — in the latter case the denominator vanishes and $\varphi=90^\circ$.

The coordinates of the intersection point of $A_1x+B_1y+C_1=0$ and $A_2x+B_2y+C_2=0$ are found by solving the system

which gives

the solution being unique exactly when $\Delta=A_1B_2-A_2B_1\neq0$. Classifying by the $2\times2$ minors of $\begin{pmatrix}A_1&B_1&C_1\\A_2&B_2&C_2\end{pmatrix}$:

• $\Delta\neq0$: the lines intersect at one point.
• $\Delta=0$ and $A_1C_2-A_2C_1,\ B_1C_2-B_2C_1$ not both zero: the lines are parallel.
• $\Delta=0$ and $A_1C_2-A_2C_1=B_1C_2-B_2C_1=0$: the lines coincide.

The signed distance (deviation) $\delta$ of a point $M(x_0,y_0)$ from the line $Ax+By+C=0$ is found by the formula

where the sign before the radical is chosen as in the normal equation (§V) — so that $\mu C\le 0$; when $C=0$, either choice is admissible and simply fixes which side of the line counts as positive. The (ordinary, nonnegative) distance $d$ from the point $M$ to the line then equals $d=|\delta|$.

The bisectors of the angles between the lines $\ell_1:\ A_1x+B_1y+C_1=0$ and $\ell_2:\ A_2x+B_2y+C_2=0$ are given by the equations

The choice of sign is made according to the figure: a point on a bisector is equidistant from both lines, $|d_1|=|d_2|$, where $d_1$ and $d_2$ are its signed distances (VIII (a)) from $\ell_1$ and $\ell_2$.

If two intersecting lines are given by the equations $A_1x+B_1y+C_1=0$ and $A_2x+B_2y+C_2=0$, then the equation

defines a line passing through the point of intersection of the given lines. Here $\lambda$ is a numerical parameter. As $\lambda$ runs over all real numbers this gives every line of the pencil except the second line $A_2x+B_2y+C_2=0$ itself (its $\lambda\to\infty$ limit); to include it, use the homogeneous form $\lambda_1(A_1x+B_1y+C_1)+\lambda_2(A_2x+B_2y+C_2)=0$ with $(\lambda_1,\lambda_2)\neq(0,0)$. The pencil's centre is the point of intersection of the given lines.