Math · Coordinate Geometry and Vectors

Straight Lines formulas for JEE

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

27 formulas3 concepts

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QuestionState the formula — Angle Between Two Lines

Reveal answer

Angle Between Two Lines

\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 0angleintersection

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All 27 Straight Lines formulas

Straight Lines Basics

11 formulas

Angle Between Two Lines

\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

\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

\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

\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 = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Distance between two points in a 2D plane.

distancepoints2d

Mid-point Formula

\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

m_1 = m_2

Relationship between the slopes of two parallel lines.

applies whenLines must be non-vertical

parallelslope

Condition for Perpendicular Lines

m_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)

\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 \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 = \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

Forms of Straight Line Equations

10 formulas

Angle Bisectors of Two Lines

\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 = 0

The standard linear equation for a straight line.

applies whenA and B are not both zero

equationgeneral

Intercept Form

\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

\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

x \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

\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 - 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)

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

applies whenNon-vertical line

equationslope-interceptx-intercept

Slope-Intercept Form (y-intercept)

y = mx + c

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

applies whenNon-vertical line

equationslope-intercepty-intercept

Two-Point Form

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

Distance from a Point to a Line

6 formulas

Distance Between Parallel Lines

d = \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

L_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

\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

\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 = \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

\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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