MathematicsCoordinate GeometryJEE Main & Advanced

Coordinate Geometry Formulas

Complete chapter-wise formula sheet for Coordinate Geometry (Mathematics) covering key equations, parameters, traps, and concept diagrams for JEE Main & Advanced.

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Chapter Directory (26 Chapters)

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Coordinate Geometry773 Qs

Straight Lines & Pair of Lines

Slope-intercept, point-slope, distance from a point, angle between lines, foot of perpendicular, and homogeneous pair of lines.

7 Formulas

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Coordinate Geometry666 Qs

Circle & System of Circles

Standard circle, general equation, tangent conditions, chord of contact, director circle, and orthogonality.

9 Formulas Diagram

Open

Coordinate Geometry358 Qs

Parabola & Conics

Standard parabola y² = 4ax, parametric form, tangent, normal in slope form, focal chords, and reflection property.

8 Formulas Diagram

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Coordinate Geometry350 Qs

Ellipse

Standard ellipse x²/a² + y²/b² = 1, eccentricity, foci, auxiliary circle, director circle, and tangent conditions.

7 Formulas Diagram

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Coordinate Geometry320 Qs

Hyperbola & Rectangular Hyperbola

Standard hyperbola x²/a² - y²/b² = 1, asymptotes, conjugate hyperbola, rectangular hyperbola xy = c².

7 Formulas Diagram

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Formulas Reference (Coordinate Geometry)

5 Chapters match

Coordinate GeometryHigh Yield (2-3 Qs in JEE)773 PYQs

Straight Lines & Pair of Lines

Slope-intercept, point-slope, distance from a point, angle between lines, foot of perpendicular, and homogeneous pair of lines.

Chapter

Distance from Point to Line

d = \frac{|a x_1 + b y_1 + c|}{\sqrt{a^2 + b^2}}

Perpendicular distance from point P(x₁, y₁) to line ax + by + c = 0.

Distance between parallel lines ax + by + c₁ = 0 and ax + by + c₂ = 0 is |c₁ - c₂| / √(a² + b²).

Angle Between Two Lines

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

Acute angle between lines with slopes m₁ and m₂.

Lines are perpendicular if m₁ m₂ = -1; parallel if m₁ = m₂.

Foot of Perpendicular & Reflection Image

\frac{x - x_1}{a} = \frac{y - y_1}{b} = -k \frac{a x_1 + b y_1 + c}{a^2 + b^2} \quad (k = 1 \text{ for Foot}, \; k = 2 \text{ for Image})

Coordinates of the projection foot and mirror reflection of point (x₁, y₁) across ax + by + c = 0.

Memory trick: k = 1 for foot (1 step), k = 2 for image (2 steps).

Angle Between Pair of Lines through Origin

a x^2 + 2hxy + b y^2 = 0 \implies \tan\theta = \frac{2\sqrt{h^2 - ab}}{|a + b|}

Joint homogeneous equation of two lines passing through the origin.

Perpendicular lines condition: a + b = 0 (coefficient of x² + coefficient of y² = 0).

General Second Degree Condition for Lines

\Delta = abc + 2fgh - af^2 - bg^2 - ch^2 = 0, \quad \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} = 0

Determinant condition for general equation ax² + 2hxy + by² + 2gx + 2fy + c = 0 to represent two straight lines.

If Δ = 0 and h² ≥ ab, equation factors into two straight lines.

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}} \quad (c_1, c_2 > 0)

Equations of internal and external angle bisectors between two intersecting lines.

With c₁, c₂ > 0: positive sign gives origin-containing bisector. If a₁a₂ + b₁b₂ > 0, it is the obtuse bisector; if < 0, it is acute.

Family of Lines & Concurrency

L_1 + \lambda L_2 = 0, \quad \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = 0

Equation of any straight line passing through intersection of L₁ = 0 and L₂ = 0, and concurrency condition for 3 lines.

Area of parallelogram bounded by y = m₁x + c₁, y = m₁x + c₂, y = m₂x + d₁, y = m₂x + d₂ is |(c₁ - c₂)(d₁ - d₂)| / |m₁ - m₂|.

Coordinate GeometryHigh Yield (2-3 Qs in JEE)666 PYQs

Circle & System of Circles

Standard circle, general equation, tangent conditions, chord of contact, director circle, and orthogonality.

Chapter

General Circle Equation & Radius

x^2 + y^2 + 2gx + 2fy + c = 0 \implies \text{Center: } (-g, -f), \quad r = \sqrt{g^2 + f^2 - c}

Standard parameters of general circle.

Always ensure the coefficients of x² and y² are 1 before identifying g, f, and c.

Tangent in Slope Form

y = mx \pm r\sqrt{1 + m^2}

Equation of tangent with slope m to circle x² + y² = r².

Condition of tangency: c² = r²(1 + m²).

Tangent at a Point (T = 0)

x x_1 + y y_1 + g(x + x_1) + f(y + y_1) + c = 0

Tangent equation at point P(x₁, y₁) lying on the circle.

Replace x² -> xx₁, y² -> yy₁, 2x -> (x + x₁), 2y -> (y + y₁).

Chord of Contact & Length of Tangent

\text{Chord of Contact: } T = 0, \quad \text{Length of Tangent: } L = \sqrt{S_1} = \sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c}

Line joining contact points of tangents drawn from external point P(x₁, y₁).

Area of quadrilateral formed by tangents and radii = rL.

Condition of Orthogonal Intersection

2 g_1 g_2 + 2 f_1 f_2 = c_1 + c_2

Condition for two circles to cut each other at right angles (tangents are perpendicular).

Both circles must have coefficients of x² and y² equal to 1.

Director Circle of a Circle

x^2 + y^2 = 2 r^2

Locus of point of intersection of two perpendicular tangents to x² + y² = r².

Concentrical circle with radius √2 times the original circle radius.

Family of Circles & Radical Axis

S_1 + \lambda S_2 = 0 \quad (\lambda \ne -1), \quad S + \lambda L = 0, \quad \text{Radical Axis: } S_1 - S_2 = 0

Locus of points from which tangents drawn to two circles are of equal length. Perpendicular to line of centers.

Radical center is the point of concurrence of radical axes of three circles taken in pairs.

Lengths of Common Tangents

L_{\text{DCT}} = \sqrt{d^2 - (r_1 - r_2)^2}, \quad L_{\text{TCT}} = \sqrt{d^2 - (r_1 + r_2)^2} \quad (d = \text{center distance})

Direct and Transverse common tangents between two separated circles.

If d > r₁ + r₂: 4 common tangents; if d = r₁ + r₂: 3; if |r₁ - r₂| < d < r₁ + r₂: 2; if d = |r₁ - r₂|: 1; if d < |r₁ - r₂|: 0.

Chord with Given Midpoint & Length of Intercept

T = S_1, \quad \text{Chord Length } = 2\sqrt{r^2 - p^2}

Equation of chord of circle S = 0 bisected at point (x₁, y₁), where p is perpendicular distance from center to chord.

T = x x₁ + y y₁ + g(x+x₁) + f(y+y₁) + c, and S₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c.

Coordinate GeometryHigh Yield (2-3 Qs in JEE)358 PYQs

Parabola & Conics

Standard parabola y² = 4ax, parametric form, tangent, normal in slope form, focal chords, and reflection property.

Chapter

Standard Parabola Parameters

y^2 = 4ax \implies \text{Focus: } (a, 0), \quad \text{Directrix: } x = -a, \quad \text{Latus Rectum: } 4a

Standard horizontal parabola opened to the right with vertex at origin.

Parametric coordinates are (at², 2at) with parameter t.

Tangent in Slope Form

y = mx + \frac{a}{m} \quad (m \ne 0)

Equation of tangent to y² = 4ax with slope m.

Point of contact is (a/m², 2a/m).

Normal in Slope Form

y = mx - 2am - am^3

Equation of normal with slope m to y² = 4ax.

Three normals can be drawn from a point (h, k) to a parabola. Sum of slopes m₁ + m₂ + m₃ = 0.

Focal Chord Property

t_1 t_2 = -1, \quad \text{Length of Focal Chord} = a\left(t + \frac{1}{t}\right)^2 = 4a\csc^2\theta

Parametric relation connecting endpoints of a chord passing through the focus (a, 0).

Minimum length of any focal chord is the latus rectum (4a) at θ = 90°.

Tangent at Parameter t

ty = x + at^2

Parametric tangent to parabola at point P(at², 2at).

Intersection of tangents at t₁ and t₂ is (a t₁ t₂, a(t₁ + t₂)).

Normal at Parameter t & Re-intersection

y + tx = 2at + at^3, \quad t_2 = -t_1 - \frac{2}{t_1}

Parametric normal at (at₁², 2at₁) intersects the parabola again at parameter t₂.

Intersection of normals at t₁ and t₂ is (2a + a(t₁² + t₂² + t₁t₂), -at₁t₂(t₁ + t₂)).

Co-Normal Points & Centroid Properties

m_1 + m_2 + m_3 = 0, \quad y_1 + y_2 + y_3 = 0, \quad x_1 + x_2 + x_3 = 2(h - 2a)

Properties of the 3 points whose normals concur at (h, k).

Centroid of the triangle formed by 3 co-normal points has y-coordinate 0 (always lies on the axis of parabola).

Director Circle of Parabola

x = -a \quad (\text{The Directrix})

Locus of point of intersection of two mutually perpendicular tangents to parabola y² = 4ax.

Tangents drawn from any point on the directrix always meet at right angles (90°).

Coordinate GeometryHigh Yield (2-3 Qs in JEE)350 PYQs

Ellipse

Standard ellipse x²/a² + y²/b² = 1, eccentricity, foci, auxiliary circle, director circle, and tangent conditions.

Chapter

Eccentricity & Foci Parameters

e = \sqrt{1 - \frac{b^2}{a^2}}, \quad \text{Foci: } (\pm ae, 0), \quad \text{Directrices: } x = \pm \frac{a}{e} \quad (a > b)

Fundamental eccentricity relation for horizontal ellipse.

Latus rectum length = 2b²/a.

Focal Distance Property (SP + S'P)

SP + S'P = 2a

The sum of distances from any point on the ellipse to both foci is constant and equals the major axis length 2a.

Commonly used to identify ellipses from locus equations |z - z₁| + |z - z₂| = 2a.

Tangent in Slope Form

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

Condition of tangency: c² = a² m² + b².

Point of contact is (∓ a²m / √(a²m² + b²), ± b² / √(a²m² + b²)).

Director Circle of Ellipse

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

Locus of point of intersection of two mutually perpendicular tangents to the ellipse.

Radius of director circle is √(a² + b²).

Parametric Coordinates & Tangent

P(a\cos\theta, b\sin\theta), \quad \text{Tangent: } \frac{x\cos\theta}{a} + \frac{y\sin\theta}{b} = 1

Parametric eccentric angle θ representation of points and tangent line.

Eccentric angle θ is the angle made by corresponding point on the auxiliary circle with the major axis.

Normal in Slope & Parametric Forms

y = mx \pm \frac{m(a^2 - b^2)}{\sqrt{a^2 + b^2 m^2}}, \quad ax\sec\theta - by\csc\theta = a^2 - b^2

Equation of normal with slope m, and parametric normal at eccentric angle θ.

Maximum 4 normals can be drawn from an arbitrary point (h, k) to an ellipse.

Auxiliary Circle & Product of Focal Perpendiculars

x^2 + y^2 = a^2, \quad p_1 \cdot p_2 = b^2

The product of lengths of perpendiculars drawn from the two foci (±ae, 0) to any tangent is constant and equals b².

Feet of perpendiculars drawn from the foci to any tangent always lie on the auxiliary circle x² + y² = a².

Coordinate GeometryHigh Yield (2 Qs in JEE)320 PYQs

Hyperbola & Rectangular Hyperbola

Standard hyperbola x²/a² - y²/b² = 1, asymptotes, conjugate hyperbola, rectangular hyperbola xy = c².

Chapter

Standard Hyperbola Parameters

e = \sqrt{1 + \frac{b^2}{a^2}}, \quad \text{Foci: } (\pm ae, 0), \quad \text{Directrices: } x = \pm \frac{a}{e}

Standard horizontal hyperbola opening along x-axis.

Eccentricity e is strictly greater than 1.

Focal Distance Property (|SP - S'P|)

|SP - S'P| = 2a

Difference of distances from any point on hyperbola to the two foci is constant 2a.

Locus definition: |z - z₁| - |z - z₂| = 2a represents hyperbola.

Tangent in Slope Form

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

Condition of tangency: c² = a² m² - b².

Real tangents exist only when m² > b²/a².

Asymptotes of Hyperbola

y = \pm \frac{b}{a} x \iff \frac{x^2}{a^2} - \frac{y^2}{b^2} = 0

Straight lines passing through the origin asymptotic to the hyperbola branches.

Angle between asymptotes is 2 tan⁻¹(b/a) = 2 sec⁻¹(e).

Rectangular Hyperbola (xy = c²)

xy = c^2, \quad \text{Parametric: } \left(ct, \frac{c}{t}\right), \quad e = \sqrt{2}

Asymptotes are mutually perpendicular coordinate axes.

Tangent at parameter t is x + y t² = 2ct; normal is y - c/t = t²(x - ct) or x t³ - y t - ct⁴ + c = 0.

Conjugate Hyperbola & Eccentricity Relation

-\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \implies \frac{1}{e_1^2} + \frac{1}{e_2^2} = 1

Hyperbola whose transverse and conjugate axes are interchanged. Foci lie on y-axis at (0, ±be₂).

The hyperbola, its asymptotes, and conjugate hyperbola differ only by constant terms in their joint equations.

Normal in Slope & Parametric Forms

y = mx \mp \frac{m(a^2 + b^2)}{\sqrt{a^2 - b^2 m^2}}, \quad ax\cos\theta + by\cot\theta = a^2 + b^2

Normal to hyperbola with slope m and at parametric point (a sec θ, b tan θ).

Director Circle of Hyperbola is x² + y² = a² - b². Real director circle exists only if a > b.