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Conics formulas

12 formulas and 12 common traps for NDA Mathematics Conics, grouped by subtopic.

Full notes with worked examples

Conic Sections — Identification & Eccentricity

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What a Conic Is — Focus, Directrix, Eccentricity

Focus–directrix definition

PFPM=e\dfrac{PF}{PM} = e

Eccentricity Values & Comparing Conics

Eccentricities

ellipse e=1−b2a2,hyperbola e=1+b2a2\text{ellipse } e=\sqrt{1-\tfrac{b^2}{a^2}}, \quad \text{hyperbola } e=\sqrt{1+\tfrac{b^2}{a^2}}

Identifying a General Second-Degree Equation

Complete the square to classify

Ax2+Cy2+Dx+Ey+F=0 →complete squares standard formAx^2 + Cy^2 + Dx + Ey + F = 0 \ \xrightarrow{\text{complete squares}}\ \text{standard form}

Common traps

Ellipse and hyperbola use OPPOSITE sign relations

For the ellipse, b2=a2(1−e2)b^2 = a^2(1 - e^2) (so e<1e<1); for the hyperbola, b2=a2(e2−1)b^2 = a^2(e^2 - 1) (so e>1e>1). Using the ellipse's relation on a hyperbola — or vice versa — gives an impossible (negative) b2b^2 and the wrong eccentricity.

A second-degree equation is not always a curve

After completing the square, check the right-hand side: =0=0 can mean a point or a pair of lines, and a negative value means no real points at all. Don't assume every Ax2+Cy2+⋯=0Ax^2+Cy^2+\cdots=0 is an ellipse or hyperbola.

Parabola — Equation, Properties & Latus Rectum

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Standard Forms & Their Elements

Standard parabola

y2=4ax: focus (a,0), directrix x=−ay^2 = 4ax: \ \text{focus } (a,0), \ \text{directrix } x = -a

The Latus Rectum

Length of latus rectum

LR=4a,endpoints (a,±2a)\text{LR} = 4a, \quad \text{endpoints } (a, \pm 2a)

Focal Distance & Focal Chords

Focal distance

focal distance of (x1,y1) on y2=4ax=x1+a\text{focal distance of } (x_1,y_1) \text{ on } y^2=4ax = x_1 + a

Tangents & Chords of a Parabola

Tangent of slope m

y=mx+am(to y2=4ax)y = mx + \dfrac{a}{m} \quad (\text{to } y^2 = 4ax)

Common traps

The sign of the linear term sets the direction

x2=−3yx^2 = -3y opens DOWNWARD (negative coefficient), so the focus is below the vertex and the directrix above. Reading it as upward flips both — the most common parabola error.

Directrix of y2=4axy^2=4ax is x=−ax=-a, on the OTHER side of the vertex

The focus is at (a,0)(a,0) and the directrix is x=−ax = -a — the directrix sits on the opposite side of the vertex from the focus. Writing the directrix as x=ax = a (the focus's coordinate) is a frequent slip.

Parabola latus rectum is 4a4a, not 2a2a

The full chord through the focus is 4a4a long — its half-length (focus to one endpoint) is 2a2a. Quoting 2a2a as the latus rectum halves the answer. Read 4a4a straight off the coefficient: for y2=12xy^2 = 12x, 4a=124a = 12, so the latus rectum is 1212.

Focal distance is x1+ax_1 + a, not x1−ax_1 - a

By the focus–directrix property, the distance to the focus equals the distance to the directrix x=−ax=-a, which is x1+ax_1 + a (you ADD aa). Using x1−ax_1 - a (distance to the focus's x-coordinate) is wrong because the directrix, not the focus, sets the measurement.

Ellipse — Foci, Eccentricity & Focal Distances

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Standard Form, Foci & Eccentricity

Foci & eccentricity

c2=a2−b2,e=cac^2 = a^2 - b^2, \qquad e = \dfrac{c}{a}

The Sum of Focal Distances

Constant focal sum

PF1+PF2=2a,latus rectum=2b2aPF_1 + PF_2 = 2a, \qquad \text{latus rectum} = \dfrac{2b^2}{a}

Building the Ellipse from Given Data

Key relations

c=ae,b2=a2−c2,LR=2b2ac = ae, \quad b^2 = a^2 - c^2, \quad \text{LR} = \dfrac{2b^2}{a}

Common traps

Major axis = larger denominator

If the bigger number is under y2y^2, the major axis is vertical and the foci are on the y-axis — and c2=(larger)−(smaller)c^2 = (\text{larger}) - (\text{smaller}) always. Assuming xx is the major axis when b>ab>a puts the foci in the wrong place.

Constant focal sum is 2a2a (major axis), not 2b2b

PF1+PF2=2aPF_1 + PF_2 = 2a uses the SEMI-MAJOR axis aa (the larger denominator), giving the full major-axis length. Using 2b2b (the minor axis) or the value aa itself gives the wrong constant. For x216+y27=1\tfrac{x^2}{16}+\tfrac{y^2}{7}=1, a=4a=4, so the sum is 88.

Ellipse latus rectum is 2b2a\dfrac{2b^2}{a} — semi-MINOR squared over semi-major

The latus rectum is 2b2a\dfrac{2b^2}{a}: the SMALLER axis squared on top, the LARGER axis on the bottom. Flipping it to 2a2b\dfrac{2a^2}{b} makes it longer than the major axis, which is impossible for an ellipse.

Hyperbola — Foci & Eccentricity

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Standard Form, Foci & Eccentricity

Hyperbola foci & eccentricity

c2=a2+b2,e=ca>1c^2 = a^2 + b^2, \qquad e = \dfrac{c}{a} > 1

Parametric Form & θ-Independent Properties

Parametric identity

sec⁡2θ−tan⁡2θ=1\sec^2\theta - \tan^2\theta = 1

Common traps

Hyperbola uses PLUS: c² = a² + b²

The ellipse has c2=a2−b2c^2 = a^2 - b^2; the hyperbola has c2=a2+b2c^2 = a^2 + b^2. Carrying the ellipse's minus sign into a hyperbola is the single most common slip in this subtopic.

A hyperbola's eccentricity is always greater than 1

Since c2=a2+b2>a2c^2 = a^2 + b^2 > a^2, we have c>ac > a and so e=ca>1e = \dfrac{c}{a} > 1 for every hyperbola. An eccentricity computed as <1<1 means you used the ellipse relation c2=a2−b2c^2 = a^2 - b^2 by mistake.

(asec⁡θ,btan⁡θ)(a\sec\theta, b\tan\theta) traces a HYPERBOLA, not an ellipse

The eliminating identity is sec⁡2θ−tan⁡2θ=1\sec^2\theta - \tan^2\theta = 1, which gives x2a2−y2b2=1\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 — a hyperbola. The ellipse uses (acos⁡θ,bsin⁡θ)(a\cos\theta, b\sin\theta) with cos⁡2θ+sin⁡2θ=1\cos^2\theta + \sin^2\theta = 1. Confusing the two parametrisations flips the conic.

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