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Formula Sheet

Conic Sections Formulas

All key formulas grouped by subtopic. Each one has a quick reminder and common mistakes to watch for.

10 formulas · 7 subtopics

Standard Parabola y² = 4ax

#1
y2=4ax,Focus: (a,0),Directrix: x=−a,LR=4ay^2 = 4ax, \quad \text{Focus: } (a, 0), \quad \text{Directrix: } x = -a, \quad \text{LR} = 4a

💡 Vertex at origin. Axis along x-axis. For y² = -4ax, the parabola opens leftward. Latus rectum (LR) passes through the focus, perpendicular to the axis.

Standard Ellipse

#2
x2a2+y2b2=1,e=1−b2a2,Foci: (±ae,0)\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad e = \sqrt{1 - \frac{b^2}{a^2}}, \quad \text{Foci: } (\pm ae, 0)

💡 Here a > b. If b > a, the major axis is along the y-axis and eccentricity uses a²/b². For an ellipse, 0 < e < 1 always.

Standard Hyperbola

#3
x2a2−y2b2=1,e=1+b2a2,Asymptotes: y=±bax\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, \quad e = \sqrt{1 + \frac{b^2}{a^2}}, \quad \text{Asymptotes: } y = \pm \frac{b}{a}x

💡 For a hyperbola, e > 1 always. The conjugate hyperbola is x²/a² - y²/b² = -1. Relationship: b² = a²(e² - 1).

Tangent to Parabola in Slope Form

#4
y=mx+amy = mx + \frac{a}{m}

💡 This is the tangent to y² = 4ax with slope m. The point of contact is (a/m², 2a/m). For m = 0, the tangent is at infinity.

Tangent to Ellipse in Slope Form

#5
y=mx±a2m2+b2y = mx \pm \sqrt{a^2 m^2 + b^2}

💡 The condition for y = mx + c to be tangent to the ellipse is c² = a²m² + b². Two tangents exist for each slope (one on each side).

⚠ Using c² = a²m² - b² (hyperbola formula) instead of c² = a²m² + b² for ellipse.

Condition for Tangency to Conics

#6
Ellipse: c2=a2m2+b2,Hyperbola: c2=a2m2−b2\text{Ellipse: } c^2 = a^2 m^2 + b^2, \quad \text{Hyperbola: } c^2 = a^2 m^2 - b^2

💡 For parabola y² = 4ax: c = a/m. Note the sign difference between ellipse (+b²) and hyperbola (-b²).

⚠ Confusing the + and - signs between ellipse and hyperbola tangency conditions.

Focal Chord Property

#7
1SP+1SQ=2l,l=semi-latus rectum\frac{1}{SP} + \frac{1}{SQ} = \frac{2}{l}, \quad l = \text{semi-latus rectum}

💡 For parabola y² = 4ax, the semi-latus rectum l = 2a. If PQ is a focal chord with parameters t₁ and t₂, then t₁t₂ = -1.

Director Circle

#8
Ellipse: x2+y2=a2+b2,Hyperbola: x2+y2=a2−b2\text{Ellipse: } x^2 + y^2 = a^2 + b^2, \quad \text{Hyperbola: } x^2 + y^2 = a^2 - b^2

💡 The director circle is the locus of the point from which two perpendicular tangents are drawn. For hyperbola, it exists only when a > b.

Chord of Contact (T = 0)

#9
Parabola: yy1=2a(x+x1),Ellipse: xx1a2+yy1b2=1\text{Parabola: } yy_1 = 2a(x + x_1), \quad \text{Ellipse: } \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1

💡 T = 0 gives chord of contact from external point (x₁, y₁). Same equation works for tangent at a point on the curve. For pair of tangents: SS₁ = T².

Parametric Forms of Conics

#10
Parabola: (at2,2at),Ellipse: (acos⁡θ,bsin⁡θ),Hyperbola: (asec⁡θ,btan⁡θ)\text{Parabola: } (at^2, 2at), \quad \text{Ellipse: } (a\cos\theta, b\sin\theta), \quad \text{Hyperbola: } (a\sec\theta, b\tan\theta)

💡 Parametric form simplifies tangent and normal equations. For parabola, slope of tangent at parameter t is 1/t.

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