Introduction
In this study note, we will delve into the topic of Straight Lines and Pair of Straight Lines, which is a crucial part of the JEE Main Mathematics syllabus. Understanding these concepts is essential for solving various problems in coordinate geometry. We will break down complex ideas into smaller, digestible sections, and use examples to illustrate key points.
Straight Lines
1. Equation of a Straight Line
The equation of a straight line in a plane can be expressed in several forms:
a. Slope-Intercept Form
The slope-intercept form of a line's equation is given by:
$$ y = mx + c $$
where:
- ( m ) is the slope of the line
- ( c ) is the y-intercept (the point where the line crosses the y-axis)
Example: Find the equation of the line with a slope of 2 and a y-intercept of -3.
Solution: Using the slope-intercept form: $$ y = 2x - 3 $$
b. Point-Slope Form
The point-slope form of a line's equation is:
$$ y - y_1 = m(x - x_1) $$
where:
- ( (x_1, y_1) ) is a point on the line
- ( m ) is the slope of the line
Example: Find the equation of the line passing through the point (1, 2) with a slope of 3.
Solution: Using the point-slope form: $$ y - 2 = 3(x - 1) $$ Simplifying, we get: $$ y = 3x - 1 $$
c. Two-Point Form
The two-point form of a line's equation is:
$$ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) $$
where:
- ( (x_1, y_1) ) and ( (x_2, y_2) ) are two points on the line
Example: Find the equation of the line passing through the points (1, 2) and (3, 4).
Solution: Using the two-point form: $$ y - 2 = \frac{4 - 2}{3 - 1} (x - 1) $$ Simplifying, we get: $$ y - 2 = x - 1 $$ $$ y = x + 1 $$
d. General Form
The general form of a line's equation is:
$$ Ax + By + C = 0 $$
where ( A ), ( B ), and ( C ) are constants.
NoteThe general form is useful for various algebraic manipulations and solving systems of equations involving lines.
2. Slope of a Line
The slope ( m ) of a line passing through two points ( (x_1, y_1) ) and ( (x_2, y_2) ) is given by:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
TipA line parallel to the x-axis has a slope of 0, while a line parallel to the y-axis has an undefined slope.
3. Angle Between Two Lines
The angle ( \theta ) between two lines with slopes ( m_1 ) and ( m_2 ) is given by:
$$ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| $$
Common MistakeA common mistake is to forget the modulus, which ensures the angle is always positive.
4. Conditions for Parallel and Perpendicular Lines
- Two lines are parallel if their slopes are equal: ( m_1 = m_2 )
- Two lines are perpendicular if the product of their slopes is ( -1 ): ( m_1 m_2 = -1 )
Pair of Straight Lines
1. General Second-Degree Equation
A general second-degree equation in ( x ) and ( y ) represents a pair of straight lines if it can be factored into two linear factors:
$$ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 $$
2. Condition for Representing a Pair of Lines
The above equation represents a pair of straight lines if:
$$ abc + 2fgh - af^2 - bg^2 - ch^2 = 0 $$
3. Angle Between the Lines
If the equation ( ax^2 + 2hxy + by^2 = 0 ) represents a pair of lines, the angle ( \theta ) between them is given by:
$$ \tan \theta = \frac{2\sqrt{h^2 - ab}}{a + b} $$
ExampleExample: Find the angle between the lines represented by the equation ( 3x^2 + 4xy + y^2 = 0 ).
Solution: Here, ( a = 3 ), ( h = 2 ), and ( b = 1 ).
$$ \tan \theta = \frac{2\sqrt{2^2 - 3 \cdot 1}}{3 + 1} = \frac{2\sqrt{4 - 3}}{4} = \frac{2 \cdot 1}{4} = \frac{1}{2} $$
Thus, ( \theta = \tan^{-1} \left( \frac{1}{2} \right) ).
4. Bisectors of Angles Between Two Lines
The angle bisectors of the lines represented by ( ax^2 + 2hxy + by^2 = 0 ) are given by:
$$ \frac{x^2 - y^2}{a - b} = \frac{xy}{h} $$
NoteThe angle bisectors are important in various geometric constructions and proofs.
Conclusion
Understanding the equations and properties of straight lines and pairs of straight lines is fundamental for solving coordinate geometry problems in JEE Main Mathematics. Practice these concepts with various problems to gain a deeper understanding and improve problem-solving skills.
TipAlways check if a given second-degree equation can be factored into linear factors to determine if it represents a pair of straight lines.
