Math · Coordinate Geometry and Vectors

Conic Sections formulas for JEE

Every Conic Sections formula you need for JEE, grouped by concept.

FormulasRevision notes
35 formulas5 concepts
01Introduction to Conic Sections02Circles and Family of Circles03Parabola04Ellipse05Hyperbola
01

Introduction to Conic Sections

7 formulas

Position of a point w.r.t Circle

S1=x12+y12−r2S_1 = x_1^2 + y_1^2 - r^2

Expression to check if a point is inside, outside, or on the circle.

applies whenPoint (x1,y1)(x_1, y_1), circle S≡x2+y2−r2=0S \equiv x^2 + y^2 - r^2 = 0.
circlepositionjee-advanced

Discriminant of Conic Equation

Δ=abc+2fgh−af2−bg2−ch2\Delta = abc + 2fgh - af^2 - bg^2 - ch^2

The condition parameter for the general second degree equation.

applies whenIf Δ=0\Delta = 0, represents pair of lines. If Δ≠0\Delta \neq 0, a true conic.
conicdiscriminantjee-advanced

General Second Degree Equation

ax2+2hxy+by2+2gx+2fy+c=0ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0

General equation representing any conic section.

applies whenRepresents different conics based on discriminant Δ\Delta and h2−abh^2-ab.
conicgeneraljee-advanced

Angle between two lines

tan⁡θ=a−a′1+aa′\tan \theta = \frac{a - a'}{1+aa'}

Lacroix's formula for the angle between two lines.

applies whenSlopes of lines are aa and a′a'.
straight-lineanglehistorical

Distance from point to line

β−aα−b1+a2\frac{\beta - a\alpha - b}{\sqrt{1+a^2}}

Lacroix's formula for perpendicular distance.

applies whenFrom point (α,β)(\alpha, \beta) to line y=ax+by = ax + b.
straight-linedistancehistorical

Perpendicularity Condition

aa′+1=0aa' + 1 = 0

Condition for two straight lines to be perpendicular by Monge.

applies whenSlopes of lines are aa and a′a'.
straight-linehistorical

Point-Slope Form

y−y′=a(x−x′)y - y' = a(x - x')

Point-slope form of a linear equation by Monge.

straight-linehistorical
02

Circles and Family of Circles

4 formulas

Circle diameter form

(x−x1)(x−x2)+(y−y1)(y−y2)=0(x-x_1)(x-x_2) + (y-y_1)(y-y_2) = 0

Equation of a circle given the endpoints of its diameter.

applies whenEndpoints of diameter are (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2).
circlediametricjee-advanced

Circle general form

x2+y2+2gx+2fy+c=0x^2 + y^2 + 2gx + 2fy + c = 0

General algebraic equation of a circle.

applies wheng2+f2−c≥0g^2 + f^2 - c \ge 0
circlegeneraljee-advanced

Circle with centre at origin

x2+y2=r2x^2 + y^2 = r^2

Equation of a circle with centre (0, 0) and radius r.

applies whenCentre at origin.
circleorigin

Circle standard form

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

Equation of a circle with centre (h, k) and radius r.

applies whenStandard 2D cartesian coordinate system.
circlestandard
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03

Parabola

9 formulas

Latus Rectum of Parabola

4a4a

Length of the latus rectum for a standard parabola.

applies whenFor parabola y2=4axy^2 = 4ax.
parabolalatus-rectum

Focal Chord Property (Parabola)

t1t2=−1t_1 t_2 = -1

Relationship between parameters of endpoints of a focal chord.

applies whenEndpoints have parameters t1t_1 and t2t_2.
parabolafocal-chordjee-advanced

Parabola Parametric Coordinates

(at2,2at)(at^2, 2at)

Parametric form for any point on a standard parabola.

applies whenFor parabola y2=4axy^2 = 4ax.
parabolaparametricjee-advanced

Condition of Tangency (Parabola)

y=mx+amy = mx + \frac{a}{m}

Equation of a tangent line to a parabola with slope m.

applies whenFor parabola y2=4axy^2 = 4ax, m≠0m \neq 0.
parabolatangentjee-advanced

Inscribed Equilateral Triangle (Parabola)

L=8a3L = 8a\sqrt{3}

Side length of an equilateral triangle inscribed in a parabola with one vertex at the origin.

applies whenFor y2=4axy^2 = 4ax.
parabolatriangleexercisejee-advanced

Parabola standard (Right)

y2=4axy^2 = 4ax

Equation of a rightward opening parabola.

applies whenVertex at origin, axis along x-axis, a>0a > 0.
parabolastandard

Parabola standard (Down)

x2=−4ayx^2 = -4ay

Equation of a downward opening parabola.

applies whenVertex at origin, axis along negative y-axis, a>0a > 0.
parabolastandard

Parabola standard (Left)

y2=−4axy^2 = -4ax

Equation of a leftward opening parabola.

applies whenVertex at origin, axis along negative x-axis, a>0a > 0.
parabolastandard

Parabola standard (Up)

x2=4ayx^2 = 4ay

Equation of an upward opening parabola.

applies whenVertex at origin, axis along positive y-axis, a>0a > 0.
parabolastandard
04

Ellipse

8 formulas

Ellipse Parametric Coordinates

(acos⁡θ,bsin⁡θ)(a \cos \theta, b \sin \theta)

Parametric coordinates of a point on an ellipse.

applies whenEccentric angle θ\theta.
ellipseparametricjee-advanced

Ellipse focal distance relation

c2=a2−b2c^2 = a^2 - b^2

Relation between semi-major axis, semi-minor axis, and focus distance.

applies whena>ba > b.
ellipseparameters

Directrices of Ellipse

x=±aex = \pm \frac{a}{e}

Equations of the directrix lines for a standard horizontal ellipse.

applies whenHorizontal major axis.
ellipsedirectrix

Eccentricity of Ellipse

e=ca=1−b2a2e = \frac{c}{a} = \sqrt{1 - \frac{b^2}{a^2}}

Ratio of distance from centre to focus and centre to vertex.

applies whene<1e < 1.
ellipseeccentricity

Latus Rectum of Ellipse

2b2a\frac{2b^2}{a}

Length of the latus rectum of a standard ellipse.

applies whenSemi-major axis aa, semi-minor axis bb.
ellipselatus-rectum

Condition of Tangency (Ellipse)

y=mx±a2m2+b2y = mx \pm \sqrt{a^2m^2 + b^2}

Equation of a tangent to a standard ellipse with slope m.

ellipsetangentjee-advanced

Ellipse standard

x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

Equation of an ellipse with horizontal major axis.

applies whenCentre at origin, a>ba > b.
ellipsestandard

Ellipse vertical

x2b2+y2a2=1\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1

Equation of an ellipse with vertical major axis.

applies whenCentre at origin, a>ba > b.
ellipsestandard
05

Hyperbola

7 formulas

Hyperbola focal distance relation

c2=a2+b2c^2 = a^2 + b^2

Relation between semi-transverse axis, semi-conjugate axis, and focal distance.

hyperbolaparameters

Eccentricity of Hyperbola

e=ca=1+b2a2e = \frac{c}{a} = \sqrt{1 + \frac{b^2}{a^2}}

Ratio of distance from centre to focus and centre to vertex.

applies whene>1e > 1.
hyperbolaeccentricity

Latus Rectum of Hyperbola

2b2a\frac{2b^2}{a}

Length of the latus rectum for a standard hyperbola.

hyperbolalatus-rectum

Condition of Tangency (Hyperbola)

y=mx±a2m2−b2y = mx \pm \sqrt{a^2m^2 - b^2}

Equation of a tangent to a standard hyperbola with slope m.

applies when∣m∣>ba|m| > \frac{b}{a}
hyperbolatangentjee-advanced

Hyperbola standard

x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

Equation of a hyperbola with horizontal transverse axis.

applies whenCentre at origin.
hyperbolastandard

Hyperbola vertical

y2a2−x2b2=1\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1

Equation of a hyperbola with vertical transverse axis.

applies whenCentre at origin.
hyperbolastandard

Rectangular Hyperbola

xy=c2xy = c^2

Equation of an equilateral hyperbola rotated by 45 degrees.

applies whenAsymptotes are the coordinate axes.
hyperbolarectangularjee-advanced
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