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MathematicsCoordinate GeometryHigh Yield (2-3 Qs in JEE)

Straight Lines & Pair of Lines Formulas

Slope-intercept, point-slope, distance from a point, angle between lines, foot of perpendicular, and homogeneous pair of lines.

7 Formulas•773+ Questions in Bank

Core Formulas & Mathematical Identities

7
01

Distance from Point to Line

d=∣ax1+by1+c∣a2+b2d = \frac{|a x_1 + b y_1 + c|}{\sqrt{a^2 + b^2}}

Perpendicular distance from point P(x₁, y₁) to line ax + by + c = 0.

Examiner Tip: Distance between parallel lines ax + by + c₁ = 0 and ax + by + c₂ = 0 is |c₁ - c₂| / √(a² + b²).
02

Angle Between Two Lines

tan⁡θ=∣m1−m21+m1m2∣\tan\theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|

Acute angle between lines with slopes m₁ and m₂.

Examiner Tip: Lines are perpendicular if m₁ m₂ = -1; parallel if m₁ = m₂.
03

Foot of Perpendicular & Reflection Image

x−x1a=y−y1b=−kax1+by1+ca2+b2(k=1 for Foot,  k=2 for Image)\frac{x - x_1}{a} = \frac{y - y_1}{b} = -k \frac{a x_1 + b y_1 + c}{a^2 + b^2} \quad (k = 1 \text{ for Foot}, \; k = 2 \text{ for Image})

Coordinates of the projection foot and mirror reflection of point (x₁, y₁) across ax + by + c = 0.

Examiner Tip: Memory trick: k = 1 for foot (1 step), k = 2 for image (2 steps).
04

Angle Between Pair of Lines through Origin

ax2+2hxy+by2=0  ⟹  tan⁡θ=2h2−ab∣a+b∣a x^2 + 2hxy + b y^2 = 0 \implies \tan\theta = \frac{2\sqrt{h^2 - ab}}{|a + b|}

Joint homogeneous equation of two lines passing through the origin.

Examiner Tip: Perpendicular lines condition: a + b = 0 (coefficient of x² + coefficient of y² = 0).
05

General Second Degree Condition for Lines

Δ=abc+2fgh−af2−bg2−ch2=0,∣ahghbfgfc∣=0\Delta = abc + 2fgh - af^2 - bg^2 - ch^2 = 0, \quad \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} = 0

Determinant condition for general equation ax² + 2hxy + by² + 2gx + 2fy + c = 0 to represent two straight lines.

Examiner Tip: If Δ = 0 and h² ≥ ab, equation factors into two straight lines.
06

Angle Bisectors of Two Lines

a1x+b1y+c1a12+b12=±a2x+b2y+c2a22+b22(c1,c2>0)\frac{a_1 x + b_1 y + c_1}{\sqrt{a_1^2 + b_1^2}} = \pm \frac{a_2 x + b_2 y + c_2}{\sqrt{a_2^2 + b_2^2}} \quad (c_1, c_2 > 0)

Equations of internal and external angle bisectors between two intersecting lines.

Examiner Tip: With c₁, c₂ > 0: positive sign gives origin-containing bisector. If a₁a₂ + b₁b₂ > 0, it is the obtuse bisector; if < 0, it is acute.
07

Family of Lines & Concurrency

L1+λL2=0,∣a1b1c1a2b2c2a3b3c3∣=0L_1 + \lambda L_2 = 0, \quad \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = 0

Equation of any straight line passing through intersection of L₁ = 0 and L₂ = 0, and concurrency condition for 3 lines.

Examiner Tip: Area of parallelogram bounded by y = m₁x + c₁, y = m₁x + c₂, y = m₂x + d₁, y = m₂x + d₂ is |(c₁ - c₂)(d₁ - d₂)| / |m₁ - m₂|.

Top Examiner Traps & Pitfalls

Frequently tested misconceptions in Straight Lines & Pair of Lines

  • •Confusing foot of perpendicular formula with image formula (image formula has an extra factor of 2).
  • •Forgetting the absolute value when computing perpendicular distance.
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