Math · Coordinate Geometry and Vectors

Straight Lines formulas for JEE

Every Straight Lines formula you need for JEE, grouped by concept.

FormulasRevision notes
27 formulas3 concepts
01Straight Lines Basics02Forms of Straight Line Equations03Distance from a Point to a Line
01

Straight Lines Basics

11 formulas

Angle Between Two Lines

tan⁡θ=∣m2−m11+m1m2∣\tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|

Formula to find the acute angle between two intersecting lines.

applies when1 + m_1 m_2 \neq 0
angleintersection

Area of a Triangle

Δ=12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣\Delta = \frac{1}{2}|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|

Area of a triangle given the coordinates of its three vertices.

areatrianglevertices

Area of Triangle Formed by Lines

Δ=(c1−c2)22∣m1−m2∣\Delta = \frac{(c_1 - c_2)^2}{2|m_1 - m_2|}

Area formed by lines y=m1x+c1, y=m2x+c2, and the y-axis (x=0).

applies whenm_1 \neq m_2
areatriangleintersections

Concurrency Condition

∣A1B1C1A2B2C2A3B3C3∣=0\begin{vmatrix} A_1 & B_1 & C_1 \\ A_2 & B_2 & C_2 \\ A_3 & B_3 & C_3 \end{vmatrix} = 0

Determinant condition for three lines to intersect at a single point.

concurrencydeterminantjee-advanced

Distance Formula

PQ=(x2−x1)2+(y2−y1)2PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Distance between two points in a 2D plane.

distancepoints2d

Mid-point Formula

(x1+x22,y1+y22)\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

Coordinates of the exact middle point of a line segment.

midpointbisect

Condition for Parallel Lines

m1=m2m_1 = m_2

Relationship between the slopes of two parallel lines.

applies whenLines must be non-vertical
parallelslope

Condition for Perpendicular Lines

m1m2=−1m_1 m_2 = -1

Relationship between the slopes of two perpendicular lines.

applies whenLines must be non-vertical and non-horizontal
perpendicularorthogonal

Section Formula (Internal)

(mx2+nx1m+n,my2+ny1m+n)\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)

Coordinates of a point dividing a line segment internally in ratio m:n.

applies whenm+n \neq 0
sectioninternaldivision

Slope (Gradient)

m=tan⁡θm = \tan \theta

Slope of a line based on its inclination angle with the positive x-axis.

applies when\theta \neq 90^\circ
slopeinclinationgradient

Slope from Two Points

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

Slope of a line passing through two specified points.

applies whenx_1 \neq x_2
slopepoints
02

Forms of Straight Line Equations

10 formulas

Angle Bisectors of Two Lines

A1x+B1y+C1A12+B12=±A2x+B2y+C2A22+B22\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}}

Equations for the two angle bisectors between intersecting lines.

applies whenIntersecting lines
bisectoranglejee-advanced

General Form of a Line

Ax+By+C=0Ax + By + C = 0

The standard linear equation for a straight line.

applies whenA and B are not both zero
equationgeneral

Intercept Form

xa+yb=1\frac{x}{a} + \frac{y}{b} = 1

Equation of a line given both x and y intercepts.

applies whena \neq 0, b \neq 0
equationintercepts

Line at Angle from Origin

yx=m±tan⁡θ1∓mtan⁡θ\frac{y}{x} = \frac{m \pm \tan \theta}{1 \mp m \tan \theta}

Equation of lines passing through the origin making angle theta with y = mx + c.

applies whenm \tan \theta \neq \pm 1
angleoriginfamily

Normal Form

xcos⁡α+ysin⁡α=px \cos \alpha + y \sin \alpha = p

Line equation using perpendicular length p from origin and its angle alpha.

applies whenp \geq 0
normaljee-advanced

Parametric / Distance Form

x−x1cos⁡θ=y−y1sin⁡θ=r\frac{x - x_1}{\cos \theta} = \frac{y - y_1}{\sin \theta} = r

Line parameterized by directed distance r from a given point.

parametricdistancejee-advanced

Point-Slope Form

y−y0=m(x−x0)y - y_0 = m(x - x_0)

Equation of a line given one point and its slope.

applies whenNon-vertical line
equationpoint-slope

Slope-Intercept Form (x-intercept)

y=m(x−d)y = m(x - d)

Equation of a line given its slope and x-intercept.

applies whenNon-vertical line
equationslope-interceptx-intercept

Slope-Intercept Form (y-intercept)

y=mx+cy = mx + c

Equation of a line given its slope and y-intercept.

applies whenNon-vertical line
equationslope-intercepty-intercept

Two-Point Form

y−y1=y2−y1x2−x1(x−x1)y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)

Equation of a line passing through two specified points.

applies whenx_1 \neq x_2
equationtwo-point
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03

Distance from a Point to a Line

6 formulas

Distance Between Parallel Lines

d=∣C1−C2∣A2+B2d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}

Perpendicular distance between two parallel lines with identical coefficients A and B.

applies whenCoefficients A and B must match in both equations
distanceparallel

Family of Lines

L1+λL2=0L_1 + \lambda L_2 = 0

Equation of any line passing through the intersection of two lines L1=0 and L2=0.

applies whenL_1, L_2 intersect
familyintersectionjee-advanced

Foot of Perpendicular

h−x1A=k−y1B=−Ax1+By1+CA2+B2\frac{h - x_1}{A} = \frac{k - y_1}{B} = -\frac{Ax_1 + By_1 + C}{A^2 + B^2}

Coordinates (h,k) of the foot of perpendicular from a point to a line.

footperpendicularjee-advanced

Image of a Point

h−x1A=k−y1B=−2Ax1+By1+CA2+B2\frac{h - x_1}{A} = \frac{k - y_1}{B} = -2\frac{Ax_1 + By_1 + C}{A^2 + B^2}

Coordinates (h,k) of the mirror reflection of a point across a line.

imagereflectionjee-advanced

Distance of a Point from a Line

d=∣Ax1+By1+C∣A2+B2d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}

The perpendicular distance from a given point to a general line.

distancepoint-lineperpendicular

Perpendicular from Origin Identity

1p2=1a2+1b2\frac{1}{p^2} = \frac{1}{a^2} + \frac{1}{b^2}

Identity relating perpendicular length p from origin to intercepts a and b.

applies whena \neq 0, b \neq 0
perpendicularoriginintercepts
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