A straight line, usually abbreviated line, is an infinitely long figure with no width, depth, or curvature, an example of such physical objects as a straightedge, a taut string, or a ray of light. Straight Lines are spaces of dimension one. Straight line is sometimes referred as a line segment which is a segment of a line having two endpoints. In this section, we will see the field of the straight lines and different types of equations of straight line and how to solve the questions based on straight lines.
What Is a Straight Line?
A straight line can be defined as an figure with infinitely long one-dimension having no width. A line also a simplest locus in a plane A straight line is locus that the slope of a segment joining any two points on this locus, then the slope is constant . A straight line is a figure that is formed when two points $A(x_1, y_1)$ and $B(x_2,y_2)$ are connected with the shortest distance between them, and the line ends are extended to infinity. This creates a figure known as a straight line.
Types of Straight Lines
Straight lines can be of various types. Generally, the straight lines are classified based on their a inclination with the axes. Their inclination t refers to the angle they form with the x-axis or the y-axis. According to the inclination of straight lines, they are of the following types
Horizontal lines
Horizontal lines are defined as the lines which are generally parallel to X-axis and perpendicular to Y-axis and drawn horizontally. Inclination with X-axis is $0^\circ$.
Vertical line are defined as the lines which are parallel to Y-axis and perpendicular to X-axis and drawn vertically. Inclination with X-axis is $90^\circ $ or $270^\circ$ .
The lines are drawn in a inclines position or slanting form some angle other than $0^\circ, 90^\circ , 270^\circ , 360^\circ$ with the horizontal or vertical lines are called oblique or slanting lines.
General Equation of a Straight Line
Equation of lines can be written in different types and also be converted one to another form by algebraic manipulation. The general equation of a straight line is given below:
$$ax+by+c=0$$
Where x and y are variables,
a, b, and c are constants.
Equation of the Horizontal Lines
Distance between straight line and X-axis is $a$ and line is parallel to X-axis, then the equation of the line is$$y=\pm a$$
Equation of the Vertical Lines
Distance between straight line and Y-axis is $b$ and line is parallel to Y-axis, then the equation of the line is
$$x=\pm b$$
Slope of a line
The smallest angle made by a line with positive direction of X-axis measured in anticlockwise sense is $\theta$ , then the value of $\tan \theta $ is called the slope of the line . We denote slope by $m$.
$$m=\tan \theta = \frac{y_2-y_1}{x_2-x_1}$$
Angle between two lines
The angle θ between the two lines having slopes $m_1$ and $m_2$ is given by
$$\tan \theta =\pm \frac{m_1-m_2}{1+m_1m_2}$$
To find the acute angle between two lines, then
$$\tan \theta =| \frac{m_1-m_2}{1+m_1m_2}|$$
Slope of parallel lines
Parallel lines are the lines which do not intersect or
meet each other in a coordinate plane.. They are always the same distance
apart. Moreover, parallel lines is having the equal slope.
Slope of perpendicular lines
Two distinct lines x and y are perpendicular, written x⊥y , if their intersection form four right angles or angles with measure 90∘ . The slopes of the perpendicular lines x and y are negative reciprocals.
$$m_x=-\frac{1}{m_y}$$
Non- vertical lines having slopes $m_x$ and $m_y$ are perpendicular to each other if and only if
$$m_x\times m_y=-1$$
Equation of a straight line in Point - slope form
Let $l$ be the line passing through the point $A(x_1,y_1)$ and which has slope m. Let $P(x,y)$ be any point on the line $l$ other than $A$.
But the slope of the line l is $m$
$$\frac{y-y_1}{x-x_1}=m$$
$$y-y_1=m(x-x_1)$$
The equation of the line is
$$y-y_1=m(x-x_1)$$
Equation of a straight line in slope-intercept form
To find the equation of the line having slope $m$ and which makes intercept $c$ on the Y -axis.
Let $P(x,y)$ be any point on the line other B. Then slope of the line $l$ is
$$m=\frac{y-c}{x-0}$$
$$m=\frac{y-c}{x}$$
$$mx=y-c$$
$$y=mx+c$$
The equation of line having slope $m$ and which makes intercept $c$ on the Y-axis is
$$y=mx+c$$
Equation of a straight line in Two points form
$$\frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1}$$
Equation of a straight line in Double – Intercept form
$$\frac{x}{a}+\frac{y}{b}=1 ~~~~~~~~(a,b \neq 0)$$
Distance of a point from a line
1)
The
distance of the origin from the line $ax+by+c=0$ is given by
$$p=\big|\frac{c}{\sqrt{a^2+b^2}}\big|$$
2) The distance of the point $P(x_1,y_1)$ from the line $ax+by+c=0 $ is given by
$$p=\big|\frac{ax_1+by_1+c}{\sqrt{a^2+b^2}}\big|$$
3) The distance between the parallel lines $ax+by+c_1=0$ and $ax+by+c_2=0$ is given by
$$p=\big|\frac{c_1-c_2}{\sqrt{a^2+b^2}}\big|$$
Frequently Asked Questions – FAQ
1)
What is
general equation of straight line?
The general equation of a straight line is given below:
$$ax+by+c=0 $$
Where
x and y are variables,
a, b, and c are constants.
2)
What is
slope of a straight line?
The angle formed by straight line with positive direction X-axis is called inclination of the line $(\theta)$ and $\tan \theta$ is called slope of the line. We denote it by $m$.
$$m=\tan\theta$$
3)
What are
the types of straight lines?
Straight lines can be of various types. According to the alignment of straight lines, they are of the following types:
- Horizontal lines
- Vertical lines
- Obliged or inclined lines
The formula or equation of a straight line is given by
$$y=mx+c $$
Where $m$ is slope of the line
$c$ is the intercept made by line wit h Y-axis.
5)
What is
perpendicular line?
Two
distinct lines $x$ and $y$ are perpendicular, written $x\perp y$ , if
their intersection form four right angles or angles with measure $90^\circ $. The
slopes of the perpendicular lines $x$ and $y$ are negative reciprocals.
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