Co-ordinate Geometry Formula Sheet — JEE Main Mathematics

Q: What are the most important Co-ordinate Geometry formulas for JEE Main?

This Co-ordinate Geometry formula sheet covers all the high-yield Mathematics formulas, definitions and theorems you need for JEE Main, across Cartesian system of rectangular coordinates, Distance formula, Section formula, Locus and its equation, Slope of a line — each shown with the key result and, where useful, a worked example.

Q: Is this Co-ordinate Geometry formula sheet free?

Yes — the full chapter formula sheet is free to read online, no login or payment required.

Q: How should I revise Co-ordinate Geometry formulas?

Blurt the Co-ordinate Geometry formulas from memory, then check against this sheet to find your gaps — and practise a few previous-year questions on the chapter to make sure you can apply them under time pressure.

Every key Co-ordinate Geometry formula, definition and theorem for JEE Main Mathematics in one place — with common examiner traps and worked examples. Free to read; blurt from memory, then check your gaps.

Syllabus — topics coveredNTA · 20 sub-topics

Cartesian system of rectangular coordinates
Distance formula
Section formula
Locus and its equation
Slope of a line
Parallel and perpendicular lines
Intercepts of a line
Various forms of equations of a line
Intersection of lines
Angles between two lines
Conditions for concurrence of three lines
Distance of a point from a line
Coordinates of centroid, orthocenter and circumcenter of triangle
Equation of circle in standard form
General form of equation of circle
Radius and center of circle
Equation of circle with endpoints of diameter
Points of intersection of line and circle
Sections of conics
Equations of parabola, ellipse and hyperbola in standard forms

Coordinate Basics & Forms of a Straight Line

Slope of a line:

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}} = tan θ

, where

θ

is the angle the line makes with the positive x-axis. A horizontal line has slope

0

; a vertical line has

undefined

slope (

θ = 9 0^{\circ}

Forms of a straight line

Form	Equation
Slope-intercept	$y = m x + c$
Point-slope	$y - y_{1} = m (x - x_{1})$
Two-point	$y - y_{1} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}} (x - x_{1})$
Intercept	$\frac{x}{a} + \frac{y}{b} = 1$
Normal	$x cos α + y sin α = p$
General	$A x + B y + C = 0$ (slope $- A / B$ )

Point formulas

▸Distance: $(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$ .
▸Section (m:n internal): $(\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n})$ .
▸Centroid: $(\frac{x _{1} + x _{2} + x _{3}}{3}, \frac{y _{1} + y _{2} + y _{3}}{3})$ .
▸Area $= \frac{1}{2} ∣ x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) ∣$ ( $= 0$ if collinear).

★ Remember · The forms of a line

Slope-intercept

y = m x + c

;

point-slope

y - y_{1} = m (x - x_{1})

;

intercept

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

;

normal

x cos α + y sin α = p

;

general

A x + B y + C = 0

(slope

- A / B

🎯 Exam · Pick the form that fits the data

Given a point and slope

\Rightarrow

point-slope; two points

\Rightarrow

two-point; the intercepts on the axes

\Rightarrow

intercept form; the perpendicular distance p and angle

α

from the origin

\Rightarrow

normal form.

⚠️ Watch out · External division & sign of area

For

external

division in ratio m:n, use

\frac{m x _{2} - n x _{1}}{m - n}

(a minus sign). Always take the

absolute value

for area — the determinant can come out negative depending on the vertex order.

🚫 Examiner Trap · Examiner traps

(1) A VERTICAL line has UNDEFINED slope (not

0

) — slope

0

is a horizontal line. (2) Slope

= tan θ

where

θ

is measured from the positive x-axis. (3) Section formula is INTERNAL

(\frac{m x _{2} + n x _{1}}{m + n}, \dots)

; for external division use

m : - n

. (4) Triangle area

= \frac{1}{2} ∣ x_{1} (y_{2} - y_{3}) + \dots ∣

— take the absolute value; area

0 \Rightarrow

collinear.

Straight Lines — Angles, Distances & Pair of Lines

Angles & distances

▸Angle between lines: $tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}}$ .
▸ $Parallel$ : $m_{1} = m_{2}$ ; $perpendicular$ : $m_{1} m_{2} = - 1$ .
▸Point to line: $\frac{∣ A x _{1} + B y _{1} + C ∣}{A ^{2} + B ^{2}}$ ; between parallels: $\frac{∣ C _{1} - C _{2} ∣}{A ^{2} + B ^{2}}$ .
▸Which side of a line: read off the sign of $A x_{1} + B y_{1} + C$ .

★ Remember · Family of lines

Every line through the intersection of

L_{1} = 0

and

L_{2} = 0

L_{1} + λ L_{2} = 0

for some

λ

. Use this to pass a line through a fixed point of intersection without first finding that point.

🎯 Exam · Pair of straight lines

a x^{2} + 2 hxy + b y^{2} = 0

represents two lines through the origin; the angle between them is

tan θ = \frac{2 h ^{2} - ab}{∣ a + b ∣}

. They are

perpendicular

iff

a + b = 0

and

coincident

iff

h^{2} = ab

★ Remember · Angle bisectors

The two angle bisectors of

a_{1} x + b_{1} y + c_{1} = 0

and

a_{2} x + b_{2} y + c_{2} = 0

are

\frac{a _{1} x + b _{1} y + c _{1}}{a _{1}^{2} + b _{1}^{2}} = \pm \frac{a _{2} x + b _{2} y + c _{2}}{a _{2}^{2} + b _{2}^{2}}

— the

+

and

-

give the two (perpendicular) bisectors.

⚠️ Watch out · Concurrency vs intersection

Three lines are

concurrent

(meet at one point) iff the

3 \times 3

determinant of their coefficients is

0

. Two non-parallel lines always intersect; concurrency is the extra condition that a

third

passes through that point.

🚫 Examiner Trap · Examiner traps

(1) Parallel

\Rightarrow m_{1} = m_{2}

; perpendicular

\Rightarrow m_{1} m_{2} = - 1

(fails if one line is vertical — check separately). (2) Distance of a point from

A x + B y + C = 0

\frac{∣ A x _{1} + B y _{1} + C ∣}{A ^{2} + B ^{2}}

— keep the modulus. (3)

a x^{2} + 2 hxy + b y^{2} = 0

is a pair of lines through the origin; perpendicular IFF

a + b = 0

, coincident IFF

h^{2} = ab

. (4) The angle bisectors give TWO bisectors (

\pm

) — the

+

one bisects the angle containing the origin only under a sign convention.

The Circle

Circle: The locus of points at a fixed distance r (radius) from a fixed point (centre). Standard form

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

; general form

x^{2} + y^{2} + 2 gx + 2 f y + c = 0

with centre

(- g, - f)

and radius

g^{2} + f^{2} - c

Lines and a circle (S = x²+y²+2gx+2fy+c)

▸Tangent at $(x_{1}, y_{1})$ : $T = 0$ ; length of tangent from a point $= S_{1}$ .
▸ $y = m x + c$ is a tangent iff $c^{2} = r^{2} (1 + m^{2})$ (distance from centre $= r$ ).
▸Chord of contact from an external point $(x_{1}, y_{1})$ : $T = 0$ .
▸Diameter form: $(x - x_{1}) (x - x_{2}) + (y - y_{1}) (y - y_{2}) = 0$ .

🎯 Exam · Two circles

Orthogonal

(tangents perpendicular at intersection):

2 g_{1} g_{2} + 2 f_{1} f_{2} = c_{1} + c_{2}

Radical axis

(equal tangent lengths):

S_{1} - S_{2} = 0

— a line perpendicular to the line of centres.

★ Remember · Inside / on / outside

For

S = x^{2} + y^{2} + 2 gx + 2 f y + c

, a point

(x_{1}, y_{1})

lies

outside

S_{1} > 0

on

S_{1} = 0

inside

S_{1} < 0

. The tangent length

S_{1}

is real only for points outside.

⚠️ Watch out · Real circle needs g²+f²−c > 0

x^{2} + y^{2} + 2 gx + 2 f y + c = 0

is a real circle only when

g^{2} + f^{2} - c > 0

. If it equals

0

the 'circle' is a point; if negative, it is imaginary (no real points).

🚫 Examiner Trap · Examiner traps

(1) For

x^{2} + y^{2} + 2 gx + 2 f y + c = 0

the centre is

(- g, - f)

(NEGATIVE) and radius

g^{2} + f^{2} - c

— a real circle needs

g^{2} + f^{2} - c > 0

. (2) Tangency of

y = m x + c

c^{2} = r^{2} (1 + m^{2})

. (3)

T = 0

gives the tangent at a point AND the chord of contact from an external point — same equation, different context. (4) Director circle (locus of

⊥

tangents) is

x^{2} + y^{2} = 2 r^{2}

; two circles are orthogonal when

2 g_{1} g_{2} + 2 f_{1} f_{2} = c_{1} + c_{2}

The Parabola

Parabola: The locus of points equidistant from a fixed point (focus) and a fixed line (directrix) — eccentricity

e = 1

. The standard form

y^{2} = 4 a x

has vertex

(0, 0)

, focus

(a, 0)

, directrix

x = - a

and latus rectum

4 a

Elements of y² = 4ax

▸Vertex $(0, 0)$ , focus $(a, 0)$ , directrix $x = - a$ , axis $y = 0$ .
▸Latus rectum $= 4 a$ , with ends $(a, \pm 2 a)$ .
▸Parametric point $(a t^{2}, 2 a t)$ .
▸Other forms: $y^{2} = - 4 a x$ (left), $x^{2} = 4 a y$ (up), $x^{2} = - 4 a y$ (down).

★ Remember · Tangent & normal

Tangent at

(a t^{2}, 2 a t)

t y = x + a t^{2}

. Slope-form tangent:

y = m x + \frac{a}{m}

(touches for every m). Normal at 't':

y = - t x + 2 a t + a t^{3}

🎯 Exam · The reflection property

A ray travelling parallel to the axis reflects off the parabola through the

focus

(the basis of headlights and dish antennas). The tangent at the vertex is the y-axis.

⚠️ Watch out · Match the form to the orientation

Read which variable is squared:

y^{2} = 4 a x

opens along the x-axis,

x^{2} = 4 a y

along the y-axis. A negative sign flips the direction. Identify a from

4 a =

(coefficient) before locating the focus.

🚫 Examiner Trap · Examiner traps

(1) For

y^{2} = 4 a x

: focus

(a, 0)

, directrix

x = - a

, latus rectum

= 4 a

, eccentricity

e = 1

— don't confuse

4 a

with a. (2) The form tells the opening:

y^{2} = 4 a x

right,

y^{2} = - 4 a x

left,

x^{2} = 4 a y

up,

x^{2} = - 4 a y

down. (3) Parametric point is

(a t^{2}, 2 a t)

. (4) Tangent at

(a t^{2}, 2 a t)

t y = x + a t^{2}

; the slope-form tangent

y = m x + \frac{a}{m}

touches for ANY m.

The Ellipse

Ellipse:

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

with

a > b

: foci

(\pm a e, 0)

, eccentricity

e = 1 - \frac{b ^{2}}{a ^{2}}

(

0 < e < 1

). Its defining property is that the

sum of the focal distances

of any point on it is

2 a

The ellipse with its foci and axes, the standard equation, eccentricity, directrices, latus rectum, parametric form and director circle — The ellipse, its elements, and tangent/director circle.

Elements (a > b)

▸Foci $(\pm a e, 0)$ , $e = 1 - \frac{b ^{2}}{a ^{2}}$ ; directrices $x = \pm \frac{a}{e}$ .
▸Major axis $2 a$ , minor axis 2b; latus rectum $= \frac{2 b ^{2}}{a}$ .
▸Sum of focal distances $= 2 a$ (the focal property).
▸Parametric point $(a cos θ, b sin θ)$ .

★ Remember · Tangent & director circle

Tangent at

(a cos θ, b sin θ)

\frac{x cos θ}{a} + \frac{y sin θ}{b} = 1

; slope-form

y = m x \pm a^{2} m^{2} + b^{2}

. The

director circle

(locus of perpendicular tangents) is

x^{2} + y^{2} = a^{2} + b^{2}

⚠️ Watch out · Which axis is major?

If instead

b > a

, the major axis is the

y

-axis: the foci are

(0, \pm b e)

with

e = 1 - \frac{a ^{2}}{b ^{2}}

. Always compare

a^{2}

and

b^{2}

first — the larger denominator sits under the major axis.

🚫 Examiner Trap · Examiner traps

(1) Ellipse needs

a > b

for a horizontal major axis:

e = 1 - \frac{b ^{2}}{a ^{2}}

with

0 < e < 1

. (2) If

b > a

the major axis is VERTICAL — foci on the y-axis,

e = 1 - \frac{a ^{2}}{b ^{2}}

(swap the roles). (3) Defining property: sum of focal distances

= 2 a

(the WHOLE major axis). (4) Latus rectum

= \frac{2 b ^{2}}{a}

; director circle

x^{2} + y^{2} = a^{2} + b^{2}

The Hyperbola

Hyperbola:

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

: foci

(\pm a e, 0)

, eccentricity

e = 1 + \frac{b ^{2}}{a ^{2}}

(

e > 1

). Its defining property is that the

absolute difference

of the focal distances is

2 a

, and it has asymptotes

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

The hyperbola with its two branches, foci and asymptotes, the standard equation, the conjugate hyperbola and the rectangular hyperbola — The hyperbola, its asymptotes, and the rectangular form.

Elements

▸Foci $(\pm a e, 0)$ , $e = 1 + \frac{b ^{2}}{a ^{2}}$ ; directrices $x = \pm \frac{a}{e}$ .
▸Asymptotes $y = \pm \frac{b}{a} x$ ; latus rectum $= \frac{2 b ^{2}}{a}$ .
▸ $∣$ difference of focal distances $∣ = 2 a$ .
▸Tangent (slope-form): $y = m x \pm a^{2} m^{2} - b^{2}$ .

🎯 Exam · Conjugate & rectangular hyperbola

The

conjugate

hyperbola

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

shares the same asymptotes. A

rectangular

hyperbola has

a = b

(so

e = 2

and perpendicular asymptotes); in rotated form it is

x y = c^{2}

with parametric point

(c t, \frac{c}{t})

★ Remember · Ellipse vs hyperbola — the one sign

Everything mirrors the ellipse with

b^{2} \to - b^{2}

e = 1 + \frac{b ^{2}}{a ^{2}}

(so

e > 1

), tangent

y = m x \pm a^{2} m^{2} - b^{2}

, director circle

x^{2} + y^{2} = a^{2} - b^{2}

. The minus sign is the whole difference.

⚠️ Watch out · Tangent reality condition

For the hyperbola,

y = m x + c

is a tangent only if

c^{2} = a^{2} m^{2} - b^{2} \geq 0

, i.e.

m^{2} \geq \frac{b ^{2}}{a ^{2}}

. Lines with smaller slope are parallel to a direction inside the asymptotic cone and never touch.

🚫 Examiner Trap · Examiner traps

(1) Hyperbola has

e > 1

and

e = 1 + \frac{b ^{2}}{a ^{2}}

(PLUS sign — contrast the ellipse's minus). (2) Defining property: |DIFFERENCE| of focal distances

= 2 a

. (3) Asymptotes are

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

; the conjugate hyperbola

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

shares the SAME asymptotes. (4) Rectangular hyperbola (

a = b

e = 2

; in

x y = c^{2}

form the asymptotes are the axes, parametric

(c t, \frac{c}{t})

Conics — Eccentricity, Tangents & Common Properties

Conics by eccentricity

e

Conic	Eccentricity
Circle	$e = 0$
Ellipse	$0 < e < 1$
Parabola	$e = 1$
Hyperbola	$e > 1$

★ Remember · Eccentricity classifies the conic

A conic is the locus where (distance to focus)

= e \times

(distance to directrix):

e = 0

circle,

0 < e < 1

ellipse,

e = 1

parabola,

e > 1

hyperbola. Ellipse/hyperbola:

e = 1 \mp \frac{b ^{2}}{a ^{2}}

; rectangular hyperbola

e = 2

Shared techniques (write S, S₁, T)

▸Tangent at a point / chord of contact from an external point: $T = 0$ (the replace-and-halve rule).
▸Point vs conic: $S_{1} > 0$ outside, $= 0$ on, $< 0$ inside.
▸Slope tangents: $y = m x + \frac{a}{m}$ (parabola), $y = m x \pm a^{2} m^{2} \pm b^{2}$ (ellipse $+$ , hyperbola $-$ ).
▸Director circle (perpendicular tangents): $a^{2} + b^{2}$ (ellipse), $a^{2} - b^{2}$ (hyperbola), $2 r^{2}$ (circle).

🎯 Exam · The T = 0 rule

To write the tangent/chord of contact, replace in S:

x^{2} \to x x_{1}

y^{2} \to y y_{1}

x \to \frac{x + x _{1}}{2}

y \to \frac{y + y _{1}}{2}

x y \to \frac{x y _{1} + x _{1} y}{2}

, and set the result

T = 0

. The same T gives the polar/chord of contact.

★ Remember · Latus rectum at a glance

Parabola:

4 a

. Ellipse and hyperbola:

\frac{2 b ^{2}}{a}

(semi-latus rectum

\frac{b ^{2}}{a}

). The latus rectum is the focal chord perpendicular to the major/transverse axis.

🚫 Examiner Trap · Examiner traps

(1) Eccentricity sorts the conics:

e = 0

circle,

0 < e < 1

ellipse,

e = 1

parabola,

e > 1

hyperbola. (2) Focus–directrix: distance to focus

= e \times

distance to directrix. (3)

T = 0

(the 'replace-and-halve' rule:

x^{2} \to x x_{1}

x \to \frac{x + x _{1}}{2}

) gives the tangent/chord of contact on ANY conic. (4)

S_{1} > 0

outside,

= 0

on,

< 0

inside (for S written with

+ x^{2}

coefficient) — check the sign convention.

More JEE Main Mathematics formula sheets

Frequently Asked Questions

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

This Co-ordinate Geometry formula sheet covers all the high-yield Mathematics formulas, definitions and theorems you need for JEE Main, across Cartesian system of rectangular coordinates, Distance formula, Section formula, Locus and its equation, Slope of a line — each shown with the key result and, where useful, a worked example.

Is this Co-ordinate Geometry formula sheet free?

Yes — the full chapter formula sheet is free to read online, no login or payment required.

How should I revise Co-ordinate Geometry formulas?

Blurt the Co-ordinate Geometry formulas from memory, then check against this sheet to find your gaps — and practise a few previous-year questions on the chapter to make sure you can apply them under time pressure.

Also useful: all formula sheets · JEE Main previous-year papers · most important chapters.