Also read, Circle Formulas Area Formulas Lines and Angles Coordinate Geometry Conic Section Formula

# The Straight line:

Straight line definition is simple according to euclidian geometry that it is a breadthless length. Also, it is the collection of points which follows a straight path. Read straight line formula further in this article.

## Straight Line Formula:

Earlier we studied about basics of coordinate geometry. In this section we will review all the formulas as well as results related to it. To make it convenient to read we have listed all the important formula in exact manner. Here is all the straight line formula.

## 1. Distance Formula:

First let’s suppose two points P(x1, y1) and Q(x2, y2), then:

$$\left| {PQ} \right| = \sqrt {{{\left( {{x_2} – {x_1}} \right)}^2} + {{\left( {{y_2} – {y_1}} \right)}^2}}$$

## 2. Mid Point Formula:

Let’s suppose two points P(x1, y1) and Q(x2, y2), then mid point of the given line is:

$$\overline {PQ} = \left( {\frac{{{x_1} + {x_2}}}{2},\frac{{{y_1} + {y_2}}}{2}} \right)$$

## 3. Any point formula:

Case1.

When the point A(x, y) divides the line PQ internally in the ratio k1 : k2:
x = $$\frac{{{k_1}{x_2} + {k_2}{x_1}}}{{{k_1} + {k_2}}}$$, y = $$\frac{{{k_1}{y_2} + {k_2}{y_1}}}{{{k_1} + {k_2}}}$$Case2.

When the point A(x, y) divides the line PQ externally in the ratio k1 : k2:

x = $$\frac{{{k_1}{x_2} – {k_2}{x_1}}}{{{k_1} – {k_2}}}$$, y = $$\frac{{{k_1}{y_2} – {k_2}{y_1}}}{{{k_1} – {k_2}}}$$

## 4. Slope of line:

straight line formula for any line $$\overline {PQ}$$, slope is given by:
m = $$\frac{{{y_2} – {y_1}}}{{{x_2} – {x_1}}}$$Notes:

The slope of the x-axis is zero.

The slope of a line parallel to the x-axis is zero.

Also, the slope of y-axis is not defined. It is ∞.

Again, the slope of the line parallel to the y-axis is not defined. It is ∞.

## 5. Straight line results on Equation:

We write the equation of x-axis as y = 0

The equation of x-axis is y=0.

The equation of line parallel to the x-axis and situated at a distance ‘a’ is y = a.

Also, the equation of line parallel to the x-axis and situated at a distance ‘b’ is x = b.

## 6. General Equations involving straight line formula:

1. Slope-intercept form:
The equation of the line having slope m and y-intercept c is y = mx + c.
1. Point-slope form:
The equation of line having slope m and passing through (x1, y1) is y – y1 = m(x – x1).
1. Equation of line passing through two points:
$$\frac{{y – {y_1}}}{{{y_2} – {y_1}}} = \frac{{x – {x_1}}}{{{x_2} – {x_1}}}$$
1. Intercept form:
The equation of line having x-intercept ‘a’ and y-intercept ‘b’ is given as:
$$\frac{x}{a} + \frac{y}{b}$$ = 1
1. Normal Form:
The normal form of a straight line is given by the equation:
$$x\cos \alpha + y\sin \alpha = p$$
where p is the length of the perpendicular from O(0,0) to the line, and α is the inclination of the perpendicular.
2. General form:
The general form of the straight line equation is:
ax + by + c=0.

## Let’s suppose two lines l1 having slope m1 and l2 having slope m2 are:

1. If lines l1 and l2 are parallel then m1 = m2.
2. If lines l1 and l2 are perpendicular then m1.m2 = -1
3. Also, If θ is the angle between l1 and l2 then
tanθ = $$\frac{{{m_2} – {m_1}}}{{1 + {m_1}{m_2}}}$$
4. Distance of any point from the line:
The straight line formula for distance of point (x1,y1)from the line ax + by + c=0is:
$$\frac{{\left| {a{x_1} + b{y_1} + c} \right|}}{{\sqrt {{a^2} + {b^2}} }}$$

## Three lines to be concurrent:

All the Three lines a1x + b1y + c1 = 0, a2x + b2y + c2 = 0, a3x + b3y + c3 = 0 are concurrent if
$$\left| \begin{array}{ccc} {a_1} & {b_1} & {c_1} \\ {a_2} & {b_2} & {c_2} \\ {a_3} & {b_3} & {c_3} \end{array} \right| = 0$$