Family of straight lines
Family of Straight Lines
The concept of the family of straight lines is an important topic in coordinate geometry. It deals with the collection of lines that share a common geometric property. This property could be passing through a fixed point, being parallel to a given line, or satisfying a particular condition.
General Equation of a Line
Before we delve into the family of straight lines, let's recall the general equation of a straight line in the Cartesian plane:
$$ Ax + By + C = 0 $$
where $A$, $B$, and $C$ are constants. This equation represents a single straight line.
Family of Lines Passing Through the Intersection of Two Lines
Consider two lines $L_1$ and $L_2$ given by their equations:
$$ L_1: a_1x + b_1y + c_1 = 0 $$ $$ L_2: a_2x + b_2y + c_2 = 0 $$
The family of lines that pass through the intersection of $L_1$ and $L_2$ can be represented by the equation:
$$ L: a_1x + b_1y + c_1 + \lambda(a_2x + b_2y + c_2) = 0 $$
where $\lambda$ is a parameter. Every value of $\lambda$ gives a different line in the family, all of which pass through the intersection point of $L_1$ and $L_2$.
Family of Lines Parallel to a Given Line
If we have a line $L_3: Ax + By + C = 0$ and we want to find the family of lines parallel to it, all lines in this family will have the same slope as $L_3$. The general equation of this family is:
$$ Ax + By + C' = 0 $$
where $C'$ is a constant that varies for different lines in the family.
Family of Lines Perpendicular to a Given Line
For a line $L_4: Ax + By + C = 0$, the family of lines perpendicular to it will have slopes that are negative reciprocals of the slope of $L_4$. The equation of this family is:
$$ Bx - Ay + C'' = 0 $$
where $C''$ is a constant that varies for different lines in the family.
Table of Differences and Important Points
| Property | Family of Lines Through Intersection | Family of Lines Parallel | Family of Lines Perpendicular |
|---|---|---|---|
| General Equation | $a_1x + b_1y + c_1 + \lambda(a_2x + b_2y + c_2) = 0$ | $Ax + By + C' = 0$ | $Bx - Ay + C'' = 0$ |
| Parameter | $\lambda$ varies to get different lines | $C'$ varies to get different lines | $C''$ varies to get different lines |
| Common Feature | Pass through a common point | Have the same slope | Slopes are negative reciprocals |
| Example | Lines passing through the intersection of $x+y=1$ and $x-y=1$ | Lines parallel to $x+2y=3$ | Lines perpendicular to $3x+4y=12$ |
Examples
Example 1: Family of Lines Through Intersection
Given two lines $L_1: x+y=1$ and $L_2: x-y=1$, find the family of lines passing through their intersection.
The family of lines is given by:
$$ x + y - 1 + \lambda(x - y - 1) = 0 $$
For different values of $\lambda$, we get different lines in the family. For example:
- If $\lambda = 1$, the line is $2x - 2 = 0$ or $x = 1$.
- If $\lambda = -1$, the line is $2y - 2 = 0$ or $y = 1$.
Example 2: Family of Lines Parallel to a Given Line
Find the family of lines parallel to the line $3x + 4y = 12$.
The family of lines will have the form:
$$ 3x + 4y + C' = 0 $$
Different values of $C'$ will give us different parallel lines. For instance:
- If $C' = -12$, the line is $3x + 4y - 12 = 0$.
- If $C' = 12$, the line is $3x + 4y + 12 = 0$.
Example 3: Family of Lines Perpendicular to a Given Line
Find the family of lines perpendicular to the line $x + 2y = 3$.
The family of lines will have the form:
$$ 2x - y + C'' = 0 $$
Different values of $C''$ will give us different perpendicular lines. For example:
- If $C'' = 3$, the line is $2x - y + 3 = 0$.
- If $C'' = -3$, the line is $2x - y - 3 = 0$.
Understanding the family of straight lines is crucial for solving various problems in coordinate geometry, especially when dealing with loci, intersections, and properties of geometric figures.