Family simply means a collection of individuals that share common properties among them. Similarly, a family of lines is a collection of lines that share some common properties among them. These lines are related to each other via some specific characteristics, such as being parallel, perpendicular, or intersecting each other. A line is the most basic and elementary form of geometry. It has only length and no width or height.
This article talks about the family of lines, its properties, types, and equations. But before learning about the family of lines, let’s get started with lines and their properties.
What is a Line?
A line, in mathematics, is a one-dimensional figure that extends in both directions endlessly and does not have a thickness. Fig. 1.1 below represents a line. A line, often called a straight line, is represented by arrows at both ends, denoting that it extends infinitely in both directions.
Equation of a Line
The general Equation of a line is given as :
ax + by + c = 0
where, a, b and c are constants and a, b ≠0. When we represent the above equation graphically, we always get a straight line.
Slope-Intercept Form of Equation
In Slope-intercept form, equation of a line is given as :
y = mx + c
where,
- m = slope of line
- c = y-intercept of line
To understand this equation better, we must first understand the concepts of slope and intercept.
Slope of the Line
Slope of a line tells us how steeply the line rises or falls. It is thus also referred to as gradient of the line.
Slope(m) = (y2 – y1)/(x2 – x1)
where,
- (x1, y1) and (x2, y2) are coordinates any two points on the line
Intercept of the Line
The point where the line crosses the axis of the line is called intercept of the line. If a point crosses the x-axis, then it is called as x-intercept and if it crosses the y-axis, then it is called as y-intercept of the line.
Fig 1.2 shows the two intercepts of the line.
y-intercept of a Line
The y-intercept is the point where a line crosses the y-axis. Simply put, it is the value of y when x = 0.
Fig 1.2 Shows the y-intercept of the line.
Fig 1.2
What is a Family of Lines?
A family of lines, in geometry, means group of lines having common characteristics between them. The common characteristics may include slope of the lines or the intercept of the lines.
Equation of a Family of Lines
The general equation of a family of lines through the point of intersection of two given lines is
L + ƛL’ = 0
Where,
- L = 0 and L’ = 0 are the two given lines,
- ƛ is a parameter.
The resultant line L, formed by the above equation, passes through the point formed at the intersection of lines L = 0 and L’ = 0.
Types of Families of Lines
Family of lines are of the following two types:
Family of Intersecting Straight Lines
As seen from the line equation, a straight line has two important properties viz. slope and intercept. If the y-intercept of a family of lines is same, they form what is called as family of intersecting straight lines. A family of Intersecting Lines pass through a common point. The slope for each line may vary, keeping y-intercept constant for all. The general equation of this family can be given as :
y – y1 = m(x – x1)
Fig 1.3 shows the family of intersecting lines.
Fig 1.3
Family of Parallel Straight Lines
Family of Parallel Straight Lines, consist of lines whose slope remains the same for all, but the y-intercept varies.
If there is a line ax + by + c = 0, then the line parallel to this line is given by ax +by + k = 0, where k is a parameter.
Fig 1.4 shows a family of parallel straight lines graphically.
Fig 1.4
Family of Perpendicular Straight Lines
Family of perpendicular straight lines refers to a set of straight lines in a plane such that each line in the family is perpendicular to each other. We can say that, any two lines chosen from this family will always intersect at right angles.
If a family of lines is being represented as ax + by +c = 0 then, any two lines from this family having slopes m1 and m2 respectively, will be perpendicular if
m1 × m2 = -1
If there is a line ax + by + c = 0, then the line perpendicular to this line is given by
bx – ay + k = 0
where k is a parameter.
Properties of Family of Lines
We have learnt that the lines belonging to a particular family of lines shares some common property. Let’s have a look on the properties of Family of Lines:
- The member of family of lines share common properties such as all member will be parallel or intersecting or perpendicular.
- For an intersecting straight line, the sum of angles formed by adjacent crossing lines will be equal to 180°.
- The opposite angles at each intersection will be equal if the family of lines have same slope.
- In family of perpendicular lines, the angle between any two lines is 90°
- A Family of Line can have infinite members having common properties
Family of Lines Examples
Example 1: Find the equation of the line which passes through the point of intersection of x + 3y – 2 = 0 and 3x – y + 4 = 0, and whose slope is 2.
Solution:
Since we need a line that passes through the intersection of x + 3y – 2 = 0 and 3x – y + 4 = 0, the equation of the line becomes:
x + 3y – 2 + ƛ(3x – y + 4) = 0
Next we find the slope of this line,
x + 3y – 2 + ƛ(3x – y + 4) = 0
⇒ x + 3y – 2 + 3ƛx – ƛy + 4ƛ = 0
⇒ (1 + 3ƛ)x + (3 – ƛ)y + 4ƛ – 2 = 0
⇒ (3 – ƛ)y = (2 – 4ƛ) – (1 + 3ƛ)x
⇒ y = (2 – 4ƛ)/(3 – ƛ) – (1 + 3ƛ)x/(3 – ƛ)
⇒ y = -(1 + 3ƛ)x/(3 – ƛ) + (2 – 4ƛ)/(3 – ƛ)
This equation is now in form y = mx + c, where m is the slope of the line, where m= -(1+3ƛ)/(3-ƛ)
We know that slope of the line is given as 2. Thus equating the above statement to m = 2, we get,
-1 – 3ƛ = 2(3 – ƛ)
⇒ -1 – 3ƛ = 6 – 2ƛ
⇒ -3ƛ + 2ƛ = 6 + 1
⇒ -ƛ = 7
Thus, ƛ = -7
Therefore, the required line equation is,
x + 3y – 2 + ƛ(3x – y +4) = 0
⇒ x + 3y – 2 – 7(3x – y + 4) = 0
⇒ x + 3y – 2 – 21x + 7y – 28 = 0
⇒ -20x + 10y – 30 = 0
⇒ -2x + y – 3 = 0
Thus the required line equation is -2x + y – 3 = 0
Frequently Asked Questions on Family of Lines
How do you obtain family of parallel straight lines?
To obtain a family of parallel lines, keep the slope constant while varying the y-intercept of the lines.
How do you obtain family of intersecting straight lines?
To obtain family of intersecting straight lines change the slope while keeping the y-intercept constant
What is the equation of family of lines?
Equation of family of lines is given as L + ƛL’ = 0
What are concurrent lines?
Concurrent lines are the three or more lines which pass through a common point. In this case, all lines have the same y-intercept and varying slope.
What is equation of lines?
Family of lines can be represented in the following two forms:
- Standard Form: ax + by + c = 0
- Slope-intercept Form : y = mx + c
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