Ch 5Straight Line — Class 11 Mathematics, concept-first.

Before we formally define a locus, it helps to recall two shapes you already know well from geometry class, because both are secretly defined by an equidistance condition — exactly the pattern a locus…

A locus is the set of every point in a plane that satisfies one fixed geometrical condition (or a fixed combination of conditions).

Suppose every point on a locus has coordinates satisfying some algebraic equation in and , and — just as importantly — no point off the locus satisfies that equation.

Let be a chosen point in the -plane, and imagine sliding the whole coordinate system so that the origin moves to , while the new axes and stay parallel to the original axes , .

A straight line is the simplest possible locus in a plane: its defining characteristic is that the slope of the segment joining any two points on it is always the same constant value.

(This sub-section is printed as "5.1.2" a second time in the source scan — an evident numbering slip, since it falls after 5.2's intercept discussion and before 5.2.2 Slope; it is presented here at it…

If a line has inclination , then — when it exists — is called the slope of the line, denoted :

The coordinate axes themselves are perpendicular, and more generally any horizontal and any vertical line are perpendicular — in that special case one line has slope and the other has an undefined slo…

Having handled the perpendicular case, we now find the acute angle between two lines that intersect but are not perpendicular.

An equation in and that is satisfied by the coordinates of every point on a line, and by no other point, is called the equation of the line.

Goal. Find the equation of the line with slope passing through a known point .

Goal. Find the equation of the line with slope that makes Y-intercept .

Goal. Find the equation of the line through two known points and .

Goal. Find the equation of the line making non-zero intercepts (on the X-axis) and (on the Y-axis).

Goal. Let be a line, and let be the perpendicular ("normal") dropped from the origin onto , with and ray making angle with the positive X-axis. Find the equation of .

Consider any straight line in a plane; it splits every point not on the line into two groups — the points on one side, and the points on the other.

Every line's equation can be written in the form called the general form. For example, rewrites as , and rewrites as .

Theorem. The perpendicular distance of the origin from the line is

Theorem. The perpendicular distance of a point from the line is

Theorem. The distance between the parallel lines and (same , so genuinely parallel) is

A collection of lines sharing one common property is called a family of lines. For instance, the set of all lines through the origin is a family — every member has equation for some value of , and dif…