We all know or have met many types of families throughout our life. In truth, each family has some distinguishing quality that is shared by all members of that family. A family, for example, may all have similar facial features, yet another family may all be sportsmen or foodies. What do you suppose a family of lines in Mathematics would look like?
On that note, let’s learn about what constitutes a family of lines, their similar features and the General Equation of a Line in detail.
What is a Family of Lines?
A Family of Lines is a group of lines that share at least one feature.
A straight line’s equation is given by,
The two most crucial features of a straight line in this context are:
- the slope of the line
While the slope defines how much the line rises or falls, the y−intercept indicates the point at which the line and the y−axis intersect.
As a result, there are 2 types of families.
Types of Family of Lines
Based on their similar features, the family of lines is divided into two categories.
- A group of lines with the same slope
- A group of lines with the same y−intercept.
Let us first learn to write the joint equation of lines before learning to write the general equation of a family of lines.
Family of Lines with the Same Slope
This family is represented as follows.
Take note of the fact that all of these lines have the same slope. They are, in other words, parallel. Although the lines appear to be identical, they have different y-intercepts. Take note of how the y-intercepts of these lines shift along the yaxis.
The lines have smaller values of b when the y-intercepts are smaller. As the line on the vertical axis rises, so does the value of b. The vertical shift is the name given to this phenomenon.
Family of Lines with the Same y-intercept
This family can look like the one illustrated below.
Take note that all of these lines converge at (0,2). A family of lines is made up of these plus an endless number of other lines that can be drawn via a specific location. Concurrent lines are formed when three or more lines intersect at a single place. As a result, this is a family of concurrent lines.
General Equation of a line
The following are the general forms of a line equation:
- Slope-intercept form
- Intercept form
- Normal form
Let us learn all the straight line formulas along with the general equation of a line and different forms to find the equation of a straight line in detail here.
General Equation of Different Types of Lines
In Geometry, there are various varieties of lines. Geometry is created from the inspiration of lines. Straight lines are classified into three types: horizontal lines (sleeping lines), vertical lines (sleeping lines), and oblique lines (Slanting lines).
The general equation of a straight line is y = mx + c, where m is the slope and c is the y-intercept. It is the most common sort of line equation in geometry. The equation of a straight line can be expressed in a variety of forms, including point-slope form, slope-intercept form, general form, standard form, and so on.
There are three ways to write general equations: point-slope form, standard form, and slope-intercept form.
The points where a line crosses the x-axis or the y-axis are named the intercepts, given by (a, 0) and (0, b).
- Slope (m) of a non-vertical line going between (x₁, y₁ ) and (x₂, y₂) points
m=(y₂ – y₁)/(x₂ – x₁), x₁ ≠ x₂
- A horizontal line’s equation
y = a or y = -a
- A vertical line’s equation
x = b or x = -b
- Line equation running through locations (x₁, y₁) and (x₂, y₂)
y – y₁= (y₂−y₁)/(x₂−x₁)×(x – x₁)
- The normal type of line equation
x cos α+y sin α = p