Mathematics · JEE

Co-ordinate Geometry Formula Sheet for JEE

73+ JEE formulas in this unit

Quick answer

The Co-ordinate Geometry JEE formula sheet lists 73+ important formulas for JEE Main and Advanced, including essential identities from Cartesian system of rectangular coordinates in a plane, Distance formula, sections formula, locus and its equation, The slope of a line, parallel and perpendicular lines, Intercepts of a line on the co-ordinate axis, and more. Revise essential formulas first, then practise MCQs on Goodmarks.

Download-free JEE mathematics formula revision for Co-ordinate Geometry. This unit-wise formula list covers 73+ exam-relevant results across Cartesian system of rectangular coordinates in a plane, Distance formula, sections formula, locus and its equation, The slope of a line, parallel and perpendicular lines, Intercepts of a line on the co-ordinate axis, and more, organised by subtopic for quick last-minute revision.

JEE Formula Sheet

73 formulas across 11 subtopics — organised for JEE Main & Advanced revision

Practise MCQs for this unit

Essential: 40Important: 29Supplementary: 4

Cartesian system of rectangular coordinates in a plane

Cartesian coordinates of a pointEssential

P(x,y)

Variables: $x$ is abscissa, $y$ is ordinate of point $P$ .
Conditions: Rectangular Cartesian axes in a plane.
Where used in JEE: Basic plotting, identification of quadrants, coordinate geometry setup.

Quadrants by signs of coordinatesImportant

\text{I}: (x>0,y>0),\quad \text{II}: (x<0,y>0),\quad \text{III}: (x<0,y<0),\quad \text{IV}: (x>0,y<0)

Variables: $x,y$ are coordinates of a point.
Conditions: Point not lying on axes.
Where used in JEE: Sign-based geometry, graph interpretation, region problems.

Origin and coordinate axesEssential

O(0,0),\ x\text{-axis}: y=0,\ y\text{-axis}: x=0

Variables: $x,y$ are coordinates.
Where used in JEE: Equation identification, intercept and symmetry questions.

Distance formula, sections formula, locus and its equation

Distance between two pointsEssential

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

Variables: $P(x_1,y_1), Q(x_2,y_2)$ .
Where used in JEE: Length of segment, triangle geometry, circle and locus problems.

Distance from originImportant

OP=\sqrt{x^2+y^2}

Variables: $P(x,y)$ .
Where used in JEE: Radius from origin, polar-like conversions, circle with centre at origin.

Section formula for internal divisionEssential

P\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}\right)

Variables: Point $P$ divides $A(x_1,y_1)$ , $B(x_2,y_2)$ internally in ratio $m:n=AP:PB$ .
Conditions: $m,n>0$ .
Where used in JEE: Division of line segment, centroid, concurrency, locus.

Section formula for external divisionImportant

P\left(\dfrac{mx_2-nx_1}{m-n},\dfrac{my_2-ny_1}{m-n}\right)

Variables: Point $P$ divides $A(x_1,y_1)$ , $B(x_2,y_2)$ externally in ratio $m:n=AP:PB$ .
Conditions: $m\ne n$ .
Where used in JEE: External division, harmonic division, conic and line problems.

Midpoint formulaEssential

M\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)

Variables: $M$ is midpoint of $A(x_1,y_1)$ , $B(x_2,y_2)$ .
Where used in JEE: Diameter of circle, medians, triangle geometry.

Area of triangle using coordinatesImportant

\Delta=\dfrac12\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|

Variables: Vertices are $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ .
Where used in JEE: Collinearity, centroid, triangle geometry, locus.

Condition of collinearity of three pointsEssential

x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)=0

Variables: Three points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ .
Where used in JEE: Line verification, concurrency, locus.

Locus equation principleImportant

\text{Replace }P(x,y)\text{ by variable coordinates and translate the geometric condition into an equation in }x,y

Variables: $P(x,y)$ is a moving point.
Conditions: Equation obtained after eliminating parameters.
Where used in JEE: Formation of locus from distance/ratio/angle conditions.

The slope of a line, parallel and perpendicular lines

Slope of a line through two pointsEssential

m=\dfrac{y_2-y_1}{x_2-x_1}

Variables: Line through $(x_1,y_1)$ and $(x_2,y_2)$ .
Conditions: $x_2\ne x_1$ .
Where used in JEE: Equation of line, angle, parallel/perpendicular tests.

Slope in terms of inclinationEssential

m=\tan\theta

Variables: $\theta$ is inclination of the line with positive $x$ -axis.
Conditions: For non-vertical line; $\theta\ne \frac{\pi}{2}+k\pi$ .
Where used in JEE: Angle interpretation, line orientation.

Slope of horizontal and vertical linesEssential

y=c\Rightarrow m=0,\quad x=c\Rightarrow m\text{ undefined}

Variables: $c$ is a constant.
Where used in JEE: Special line cases, angle and intercept questions.

Parallel lines condition using slopeEssential

m_1=m_2

Variables: $m_1,m_2$ are slopes of two lines.
Conditions: For distinct lines, intercepts differ.
Where used in JEE: Line relation, parameter determination.

Perpendicular lines condition using slopeEssential

m_1m_2=-1

Variables: $m_1,m_2$ are slopes of two non-vertical lines.
Conditions: Equivalent special case: one line vertical and the other horizontal.
Where used in JEE: Orthogonality, altitudes, tangent-normal problems.

General line coefficients and slope relationEssential

ax+by+c=0\Rightarrow m=-\dfrac{a}{b}

Variables: $a,b,c$ are constants.
Conditions: $b\ne 0$ . If $b=0$ , line is vertical.
Where used in JEE: Converting line equations, angle and parallel/perpendicular conditions.

Intercepts of a line on the co-ordinate axis

Intercept form of a lineEssential

\dfrac{x}{a}+\dfrac{y}{b}=1

Variables: $a$ and $b$ are intercepts on $x$ -axis and $y$ -axis respectively.
Conditions: $a\ne 0, b\ne 0$ .
Where used in JEE: Intercept questions, triangle area with axes, line identification.

Intercepts from general equationImportant

ax+by+c=0\Rightarrow x\text{-intercept}=-\dfrac{c}{a},\ y\text{-intercept}=-\dfrac{c}{b}

Variables: $a,b,c$ are constants.
Conditions: Need $a\ne 0$ for $x$ -intercept and $b\ne 0$ for $y$ -intercept.
Where used in JEE: Finding intercepts directly from line equation.

Straight line: Various forms of equations of a line, intersection of lines, angles between two lines

One-point form of line through originImportant

y=mx

Variables: $m$ is slope.
Where used in JEE: Lines through origin, family of lines, conic chords through origin.

Slope-point form of lineEssential

y-y_1=m(x-x_1)

Variables: $(x_1,y_1)$ is a point on the line, $m$ is slope.
Where used in JEE: Equation from one point and slope.

Two-point form of lineEssential

\dfrac{y-y_1}{x-x_1}=\dfrac{y_2-y_1}{x_2-x_1}

Variables: Line through $(x_1,y_1)$ and $(x_2,y_2)$ .
Conditions: $x_1\ne x_2$ ; otherwise line is $x=x_1$ .
Where used in JEE: Equation from two points.

Two-intercept form of lineImportant

\dfrac{x}{a}+\dfrac{y}{b}=1

Variables: $a,b$ are x- and y-intercepts.
Conditions: $a,b\ne 0$ .
Where used in JEE: Axis intercept problems.

Normal form of lineImportant

x\cos\alpha+y\sin\alpha=p

Variables: $p$ is perpendicular distance from origin, $\alpha$ is angle made by the normal with positive $x$ -axis.
Conditions: $p\ge 0$ .
Where used in JEE: Distance from origin, tangent forms, shortest distance problems.

General form of lineEssential

ax+by+c=0

Variables: $a,b$ not both zero.
Where used in JEE: Universal line representation, line relations, distance formulas.

Parametric form of line through a pointSupplementary

x=x_1+r\cos\theta,\ y=y_1+r\sin\theta

Variables: $(x_1,y_1)$ is a fixed point, $\theta$ is inclination, $r\in\mathbb{R}$ .
Where used in JEE: Intersection, distance, line-circle problems.

Family of lines through intersection of two linesImportant

L_1+\lambda L_2=0

Variables: $L_1=0$ and $L_2=0$ are two intersecting lines; $\lambda$ is parameter.
Conditions: Represents all lines through their point of intersection.
Where used in JEE: Pair of lines, concurrency, parameter-based line problems.

Intersection point of two lines by solving equationsEssential

a_1x+b_1y+c_1=0,\ a_2x+b_2y+c_2=0\Rightarrow \text{solve simultaneously for }(x,y)

Variables: $a_i,b_i,c_i$ are constants.
Conditions: Unique intersection if $a_1b_2-a_2b_1\ne 0$ .
Where used in JEE: Point of intersection, triangle vertices, concurrency.

Angle between two lines using slopesEssential

\tan\theta=\left|\dfrac{m_2-m_1}{1+m_1m_2}\right|

Variables: $m_1,m_2$ are slopes; $\theta$ is acute angle between lines.
Conditions: If $1+m_1m_2=0$ , then $\theta=\frac{\pi}{2}$ .
Where used in JEE: Angle, bisectors, line relation questions.

Angle between two lines using coefficientsImportant

\tan\theta=\left|\dfrac{a_1b_2-a_2b_1}{a_1a_2+b_1b_2}\right|

Variables: Lines are $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ .
Conditions: If $a_1a_2+b_1b_2=0$ , lines are perpendicular.
Where used in JEE: Angle between lines without converting to slope form.

Condition for parallel lines using coefficientsEssential

a_1b_2-a_2b_1=0

Variables: Lines are $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ .
Conditions: For distinct parallel lines, additionally $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$ .
Where used in JEE: Parallelism test in general form.

Condition for perpendicular lines using coefficientsEssential

a_1a_2+b_1b_2=0

Variables: Lines are $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ .
Where used in JEE: Perpendicularity test in general form.

Angle bisectors of two linesImportant

\dfrac{a_1x+b_1y+c_1}{\sqrt{a_1^2+b_1^2}}=\pm\dfrac{a_2x+b_2y+c_2}{\sqrt{a_2^2+b_2^2}}

Variables: Two lines are $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ .
Conditions: Applicable for intersecting lines.
Where used in JEE: Angle bisector, incenter/excenter type setups.

Conditions for concurrence of three lines, the distance of a point from a line

Condition for concurrence of three linesEssential

\begin{vmatrix}a_1&b_1&c_1\\ a_2&b_2&c_2\\ a_3&b_3&c_3\end{vmatrix}=0

Variables: Lines are $a_ix+b_iy+c_i=0$ , $i=1,2,3$ .
Where used in JEE: Testing whether three lines pass through one point.

Distance of a point from a lineEssential

d=\dfrac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}

Variables: Distance from point $(x_1,y_1)$ to line $ax+by+c=0$ .
Where used in JEE: Shortest distance, tangency, region and optimization problems.

Distance between parallel linesImportant

d=\dfrac{|c_2-c_1|}{\sqrt{a^2+b^2}}

Variables: Parallel lines are $ax+by+c_1=0$ and $ax+by+c_2=0$ .
Where used in JEE: Separation of parallels, strip region, tangent pair questions.

Co-ordinate of the centroid, orthocentre and circumcentre of a triangle

Centroid of a triangleEssential

G\left(\dfrac{x_1+x_2+x_3}{3},\dfrac{y_1+y_2+y_3}{3}\right)

Variables: Vertices are $A(x_1,y_1),B(x_2,y_2),C(x_3,y_3)$ .
Where used in JEE: Median problems, triangle coordinate geometry.

Centroid divides median in ratio 2:1Important

AG:GM=2:1

Variables: $G$ is centroid, $M$ midpoint of opposite side.
Where used in JEE: Section formula applications in triangles.

Circumcentre as intersection of perpendicular bisectorsImportant

OA=OB=OC

Variables: $O$ is circumcentre of triangle $ABC$ .
Where used in JEE: Finding centre of circumcircle from equal distances.

Circumcentre for right triangleImportant

O\text{ is midpoint of hypotenuse}

Variables: $O$ is circumcentre.
Conditions: Triangle must be right-angled.
Where used in JEE: Special triangle geometry in coordinates.

Orthocentre as intersection of altitudesEssential

H=\text{intersection point of the three altitudes}

Variables: $H$ is orthocentre.
Where used in JEE: Altitude equations, Euler line questions.

Orthocentre for right triangleImportant

H\text{ is the right-angled vertex}

Variables: $H$ is orthocentre.
Conditions: Triangle must be right-angled.
Where used in JEE: Special case simplification.

Euler line relationImportant

\vec{OG}:\vec{GH}=1:2

Variables: $O,G,H$ are circumcentre, centroid, orthocentre respectively.
Conditions: For any non-equilateral triangle, points are collinear.
Where used in JEE: Finding one triangle centre from the other two.

Coordinate relation from Euler lineImportant

H(3x_G-2x_O,\ 3y_G-2y_O)

Variables: $(x_O,y_O),(x_G,y_G)$ are coordinates of circumcentre and centroid.
Where used in JEE: Direct computation of orthocentre.

Coordinate relation for circumcentre from centroid and orthocentreSupplementary

O\left(\dfrac{3x_G-x_H}{2},\dfrac{3y_G-y_H}{2}\right)

Variables: $(x_G,y_G),(x_H,y_H)$ are coordinates of centroid and orthocentre.
Where used in JEE: Reverse Euler line problems.

Circle, conic sections: A standard form of equations of a circle, the general form of the equation of a circle, its radius and centre

Standard equation of circle with centre at originEssential

x^2+y^2=a^2

Variables: $a$ is radius.
Conditions: $a>0$ .
Where used in JEE: Basic circle geometry, line-circle intersection, tangency.

Standard equation of circle with arbitrary centreEssential

(x-h)^2+(y-k)^2=r^2

Variables: $(h,k)$ is centre, $r$ is radius.
Conditions: $r>0$ .
Where used in JEE: Centre-radius identification, tangent and chord problems.

General equation of a circleEssential

x^2+y^2+2gx+2fy+c=0

Variables: Centre is $(-g,-f)$ , radius is $\sqrt{g^2+f^2-c}$ .
Conditions: Represents a real circle if $g^2+f^2-c>0$ ; point circle if equality holds.
Where used in JEE: Conversion between general and standard forms.

Centre and radius from general circleEssential

\text{Centre}=(-g,-f),\quad r=\sqrt{g^2+f^2-c}

Variables: For circle $x^2+y^2+2gx+2fy+c=0$ .
Conditions: Real circle when $g^2+f^2-c>0$ .
Where used in JEE: Immediate extraction of centre and radius.

Circle through origin conditionSupplementary

c=0

Variables: In $x^2+y^2+2gx+2fy+c=0$ .
Where used in JEE: Checking if origin lies on circle.

Equation of a circle when the endpoints of a diameter are given

Equation of circle with diameter endpoints givenEssential

(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0

Variables: Endpoints of a diameter are $(x_1,y_1)$ and $(x_2,y_2)$ .
Where used in JEE: Circle from diameter, right angle in semicircle applications.

Centre and radius from diameter endpointsImportant

\text{Centre}=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right),\quad r=\dfrac12\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

Variables: Diameter endpoints are $(x_1,y_1),(x_2,y_2)$ .
Where used in JEE: Constructing circle from diameter data.

Points of intersection of a line and a circle with the centre at the origin

Intersection of line and circle centered at originEssential

x^2+y^2=a^2,\ y=mx+c\Rightarrow (1+m^2)x^2+2mcx+(c^2-a^2)=0

Variables: Circle radius $a$ , line slope $m$ , intercept $c$ .
Where used in JEE: Finding coordinates of intersection points, secant/tangent conditions.

Condition for line to intersect circle at origin-centred circleImportant

c^2\le a^2(1+m^2)

Variables: Line $y=mx+c$ , circle $x^2+y^2=a^2$ .
Conditions: Equality gives tangency, strict inequality gives secant.
Where used in JEE: Relative position of line and circle.

Condition for tangency of line to origin-centred circleEssential

c^2=a^2(1+m^2)

Variables: Line $y=mx+c$ , circle $x^2+y^2=a^2$ .
Where used in JEE: Tangent determination, parameter values.

Line through origin intersecting origin-centred circleImportant

y=mx\Rightarrow \left(\pm\dfrac{a}{\sqrt{1+m^2}},\ \pm\dfrac{am}{\sqrt{1+m^2}}\right)

Variables: Circle $x^2+y^2=a^2$ , line $y=mx$ .
Conditions: Signs are taken same in each coordinate.
Where used in JEE: Chord through centre, endpoints of diameter along a direction.

Sections of conics, equations of conic sections (parabola, ellipse and hyperbola) in standard forms

Standard parabola opening rightEssential

y^2=4ax

Variables: $a$ is distance of focus from vertex.
Conditions: Vertex at origin, axis along positive $x$ -axis, $a>0$ .
Where used in JEE: Basic parabola identification and property use.

Elements of parabola $y^2=4ax$Essential

\text{Vertex}=(0,0),\ \text{Focus}=(a,0),\ \text{Directrix}: x=-a,\ \text{Axis}: y=0,\ \text{Length of latus rectum}=4a

Variables: $a>0$ .
Where used in JEE: Focus-directrix, tangent/normal, chord problems.

Standard parabola opening leftImportant

y^2=-4ax

Variables: $a>0$ .
Conditions: Vertex at origin, axis along negative $x$ -axis.
Where used in JEE: Parabola orientation questions.

Standard parabola opening upward/downwardImportant

x^2=4ay

\ or

x^2=-4ay

Variables: $a>0$ .
Conditions: Vertex at origin, axis along $y$ -axis.
Where used in JEE: Vertical-axis parabola problems.

Parametric point on parabolaImportant

y^2=4ax\Rightarrow (at^2,2at)

Variables: $t\in\mathbb{R}$ is parameter.
Where used in JEE: Chord, tangent, normal, focal property problems.

Standard ellipse with major axis on x-axisEssential

\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1

Variables: $a>b>0$ ; major axis length $2a$ , minor axis length $2b$ .
Conditions: Centre at origin.
Where used in JEE: Basic ellipse identification and properties.

Ellipse focal relationEssential

c^2=a^2-b^2,\quad e=\dfrac{c}{a}

Variables: $c$ is focal distance, $e$ is eccentricity.
Conditions: For ellipse $a>b>0$ , hence $0<e<1$ .
Where used in JEE: Finding foci, eccentricity, directrices.

Elements of ellipse on x-axisEssential

\text{Centre}=(0,0),\ \text{Vertices}=(\pm a,0),\ \text{Co-vertices}=(0,\pm b),\ \text{Foci}=(\pm c,0),\ \text{Directrices}: x=\pm\dfrac{a}{e},\ \text{Length of latus rectum}=\dfrac{2b^2}{a}

Variables: For $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ , $c^2=a^2-b^2$ .
Where used in JEE: Ellipse geometry and standard results.

Standard ellipse with major axis on y-axisImportant

\dfrac{x^2}{b^2}+\dfrac{y^2}{a^2}=1

Variables: $a>b>0$ .
Conditions: Centre at origin.
Where used in JEE: Orientation-based ellipse questions.

Parametric point on ellipseImportant

\left(a\cos\theta,b\sin\theta\right)

Variables: $\theta$ is parameter.
Conditions: For ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ .
Where used in JEE: Tangent, chord, focal property, maxima-minima.

Standard hyperbola with transverse axis on x-axisEssential

\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1

Variables: $a,b>0$ .
Conditions: Centre at origin.
Where used in JEE: Basic hyperbola identification and property use.

Hyperbola focal relationEssential

c^2=a^2+b^2,\quad e=\dfrac{c}{a}

Variables: $c$ is focal distance, $e$ is eccentricity.
Conditions: For hyperbola, $e>1$ .
Where used in JEE: Foci, eccentricity, directrices.

Elements of hyperbola on x-axisEssential

\text{Centre}=(0,0),\ \text{Vertices}=(\pm a,0),\ \text{Foci}=(\pm c,0),\ \text{Directrices}: x=\pm\dfrac{a}{e},\ \text{Length of latus rectum}=\dfrac{2b^2}{a}

Variables: For $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ , $c^2=a^2+b^2$ .
Where used in JEE: Hyperbola geometry, focus-directrix property.

Asymptotes of standard hyperbolaEssential

y=\pm\dfrac{b}{a}x

Variables: For hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ .
Where used in JEE: Asymptote-based questions, rectangular hyperbola transforms.

Standard hyperbola with transverse axis on y-axisImportant

\dfrac{y^2}{a^2}-\dfrac{x^2}{b^2}=1

Variables: $a,b>0$ .
Conditions: Centre at origin.
Where used in JEE: Orientation-based hyperbola problems.

Parametric point on hyperbolaSupplementary

(a\sec\theta,b\tan\theta)

Variables: $\theta$ is parameter.
Conditions: For hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ .
Where used in JEE: Tangent, chord, asymptote-related problems.

Eccentricity classification of conicsImportant

e=1\Rightarrow \text{parabola},\quad 0<e<1\Rightarrow \text{ellipse},\quad e>1\Rightarrow \text{hyperbola}

Variables: $e$ is eccentricity.
Where used in JEE: Conic identification, focus-directrix based questions.

Frequently asked questions

What are the important Co-ordinate Geometry formulas for JEE?

This page lists 73+ JEE-relevant Co-ordinate Geometry formulas organised by subtopic. Start with essential formulas, then important identities before supplementary shortcuts.

Is this Co-ordinate Geometry formula sheet aligned with JEE Main?

Yes. Every formula is mapped to the JEE Main Mathematics syllabus for Co-ordinate Geometry, covering Cartesian system of rectangular coordinates in a plane, Distance formula, sections formula, locus and its equation, The slope of a line, parallel and perpendicular lines, Intercepts of a line on the co-ordinate axis, and more.

How should I revise the Co-ordinate Geometry formula sheet before JEE?

Revise essential formulas daily, important ones every 2–3 days, and supplementary results weekly. After each pass, solve 10–15 MCQs to test recall under exam conditions.

Where can I practise Co-ordinate Geometry MCQs after revising formulas?

Use the Online Practice or MCQs pages for the same unit on Goodmarks to convert formula recall into problem-solving speed.

Does this replace NCERT for Co-ordinate Geometry?

No — use this formula sheet for quick revision alongside NCERT and your coaching notes. Formulas here are a condensed reference, not a substitute for concept building.