Straight Lines – JEE Mains 2026 Revision Series
Straight lines often look simple, but JEE loves hiding sneaky traps inside them. Here’s a crisp and powerful revision of all formulas & concepts you need to score full marks.
1. Basics
The general equation of a straight line is:
\\( Ax + By + C = 0 \\)
2. Slope Concepts
Slope (\\(m\\)) represents inclination.
From angle of inclination: \\( m = \tan\theta \\)
From two points: \\( m = \frac{y_2 - y_1}{x_2 - x_1} \\)
From general form: \\( m = -\frac{A}{B} \\)
Parallel lines: \\( m_1 = m_2 \\)
Perpendicular: \\( m_1 m_2 = -1 \\)
3. Forms of Line Equations
Slope–Intercept: \\( y = mx + c \\)
Point–Slope: \\( y - y_1 = m(x - x_1) \\)
Two-Point: \\( \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1} \\)
Intercept Form: \\( \frac{x}{a} + \frac{y}{b} = 1 \\)
Normal Form: \\( x\cos\alpha + y\sin\alpha = p \\)
Parametric:
\\( x = x_1 + r\cos\theta \\),
\\( y = y_1 + r\sin\theta \\)
4. Distance of a Point from a Line
\\( d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \\)
5. Angle Between Two Lines
Using slopes:
\\( \tan\theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right| \\)
Using general form:
\\( \cos\theta = \frac{A_1A_2 + B_1B_2}{\sqrt{A_1^2+B_1^2}\sqrt{A_2^2+B_2^2}} \\)
6. Pair of Lines
Equation: \\( ax^2 + 2hxy + by^2 = 0 \\)
Represents two lines if: \\( h^2 = ab \\)
Angle: \\( \tan\theta = \frac{2\sqrt{h^2 - ab}}{a + b} \\)
7. Area of Triangle
\\( \Delta = \frac{1}{2}|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \\)
8. Foot of Perpendicular
From point \\( P(x_1,y_1) \\) to \\( Ax+By+C=0 \\):
\\( x = x_1 - \frac{A(Ax_1+By_1+C)}{A^2+B^2} \\)
\\( y = y_1 - \frac{B(Ax_1+By_1+C)}{A^2+B^2} \\)
9. Family of Lines
Through intersection of two lines:
\\( L_1 + \lambda L_2 = 0 \\)
Angle bisectors:
\\( \frac{L_1}{\sqrt{A_1^2+B_1^2}} = \pm \frac{L_2}{\sqrt{A_2^2+B_2^2}} \\)
10. JEE Traps
- Vertical line → slope undefined
- Horizontal line → slope 0
- Parallel to x-axis: \\( y = k \\)
- Parallel to y-axis: \\( x = k \\)
- Collinearity → area = 0
- Minimum distance exists only for parallel lines
11. JEE PYQ Patterns
- Distance between parallel lines
- Foot of perpendicular
- Locus problems involving line constraints
- Area of triangle + collinearity
- Intersection of a line with a circle
- Lines dividing a segment in ratio \\(m:n\\)
Mini Illustration (Quick JEE-Level Question)
Q. Find the distance between lines \\( 3x - 4y + 7 = 0 \\) and \\( 3x - 4y - 5 = 0 \\)
Ans. They are parallel. Use formula: \\( d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} \\)
\\( d = \frac{|7 + 5|}{\sqrt{3^2 + (-4)^2}} = \frac{12}{5} \\)
Distance = 2.4 units
Comments