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Conic Sections Notes, Formulas, Traps & PYQs | ExamiFly | Examifly

Home Chapters MathematicsConic Sections - Parabola, Ellipse and Hyperbola

MathematicsJEE Main & Advanced

2-3 Questions | 8-12 Marks in JEE Main & Advanced

Mastering Conic Sections: Parabola, Ellipse & Hyperbola | Formulas, Traps & Strategy

Q: How to quickly identify the conic section represented by $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$?

Compute the discriminant $\Delta = B^2 - 4AC$. If $\Delta 0$, a hyperbola (provided the general non-degenerate condition $\Delta_{3\times3} \neq 0$ holds).

Prerequisites:Straight Lines, Circles, Quadratic Equations, Basic Coordinate Geometry

Conic Sections Parabola Ellipse Hyperbola ExamiFly Revision Notes and Formulas

Chapter Overview

Conic sections represent the locus of a moving point $P(x,y)$ whose distance from a fixed point (focus) bears a constant ratio (eccentricity $e$ ) to its distance from a fixed line (directrix). When $e = 1$ , the trajectory curves into a parabola; when $e < 1$ , it closes into an elegant ellipse; and when $e > 1$ , it ruptures into a hyperbola. In the IIT-JEE arena, conics is not merely about memorizing equations—it is the ultimate testing ground for geometric visualization, parametric transformations, and calculus-geometry bridges. Most candidates stumble by treating parabola, ellipse, and hyperbola as isolated silos, missing the profound unifying themes like the focal property, chord of contact, and tangent condition $c = \pm \sqrt{a^2 m^2 \pm b^2}$ . Master the unified parametric approach to solve complex locus problems with speed and precision.

Core Subtopics & Syllabus Breakdown

Standard Equations and Properties of Parabola

Ellipse Equations, Auxiliary Circles & Eccentricity

Hyperbola & Rectangular Hyperbola Properties

Tangents, Normals & Director Circles

Chord of Contact & Polar Coordinates in Conics

Key Formulas & High-Yield Equations

Tangent to Parabola $y^2 = 4ax$

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

Valid for slope

m \neq 0

. Point of contact is strictly

\left(\frac{a}{m^2}, \frac{2a}{m}\right)

Tangent to Ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

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

Applies for real tangents where

a^2 m^2 + b^2 > 0

. Point of contact is

\left(-\frac{a^2 m}{c}, \frac{b^2}{c}\right)

where

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

Eccentricity & Foci of Hyperbola

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

Strictly valid for

e > 1

. Distance between foci is

2ae

, and length of latus rectum is

\frac{2b^2}{a}

Rectangular Hyperbola $xy = c^2$

$x = ct, \quad y = \frac{c}{t}$

Valid for parametric parameter

t \neq 0

. Asymptotes are mutually perpendicular lines

x = 0

and

y = 0

Focal Property of Ellipse

$SP + S'P = 2a$

Valid for any point

P

on the ellipse

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

with

a > b

, where

S

and

S'

are the foci.

Common Mistakes & Traps to Avoid

High Error Zones

Confusing focal property of Ellipse ( $SP + S'P = 2a$ ) with Hyperbola ( $|SP - S'P| = 2a$ ), leading to sign errors in locus problems.

Correct Approach:Remember: Ellipse is bounded so focal distances add up (

SP + S'P = 2a

), while Hyperbola is unbounded so absolute distance difference is constant (

|SP - S'P| = 2a

Using tangent slope form $y = mx + \frac{a}{m}$ for shifted parabolas like $(y-k)^2 = 4a(x-h)$ without applying translation.

Correct Approach:Substitute

Y = y-k

and

X = x-h

first to get

Y = mX + \frac{a}{m}

, yielding

y-k = m(x-h) + \frac{a}{m}

Assuming conjugate hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = -1$ has eccentricity $e = \sqrt{1 + \frac{b^2}{a^2}}$ .

Correct Approach:For conjugate hyperbola

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

, eccentricity is strictly

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

, satisfying

\frac{1}{e^2} + \frac{1}{e'^2} = 1

Past Year Question (PYQ) Trends

2024Medium-Hard

Questions Asked: 3

Topics: Tangents & Normals, Concentric Conics & Locus, Asymptotes of Hyperbola

2023Medium

Questions Asked: 2

Topics: Standard Equations & Focal Properties, Intersection of Conics

2022Hard

Questions Asked: 3

Topics: Locus Problems, Parametric Equations, Eccentricity Relations

Solved High-Yield Sample Questions

Practice Mode

Question #1JEE Main 2024 (Fundamental)

Find the value of $k$ for which the line $y = 2x + k$ is tangent to the parabola $y^2 = 8x$ .

Solution Locked

Step-by-Step Solution & Concept

Step 1

Compare $y^2 = 8x$ with standard parabola $y^2 = 4ax$ to get $4a = 8 \implies a = 2$ .

Step 2

Use condition of tangency for $y = mx + c$ on $y^2 = 4ax$ , which is $c = \frac{a}{m}$ .

Step 3

Here slope $m = 2$ and $c = k$ , so $k = \frac{2}{2} = 1$ .

Select an option & click Check Answer to reveal solution

Question #2JEE Main 2023

If the eccentricity of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is $\frac{5}{3}$ and the distance between its foci is $10$ , then the length of its latus rectum is:

Solution Locked

Step-by-Step Solution & Concept

Step 1

Distance between foci $2ae = 10 \implies ae = 5$ . Given $e = \frac{5}{3}$ , we get $a = 3$ .

Step 2

Use hyperbola relation $b^2 = a^2(e^2 - 1) = 3^2 \left(\left(\frac{5}{3}\right)^2 - 1\right) = 9 \left(\frac{16}{9}\right) = 16$ .

Step 3

Length of latus rectum is $\frac{2b^2}{a} = \frac{2(16)}{3} = \frac{32}{3}$ .

Select an option & click Check Answer to reveal solution

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Frequently Asked Questions

How do I decide whether to use

y = mx + c

form or parametric form

(a t^2, 2at)

for parabola problems?

Use slope form $y = mx + \frac{a}{m}$ when line parameters or angles with axes are given. Use parametric coordinates $(a t^2, 2at)$ when dealing with chords, intersection of tangents at two specific points, or calculus-based optimizations.

What is the relation between eccentricities of a hyperbola and its conjugate hyperbola?

If $e_1$ is the eccentricity of hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $e_2$ is the eccentricity of its conjugate $\frac{x^2}{a^2} - \frac{y^2}{b^2} = -1$ , then $\frac{1}{e_1^2} + \frac{1}{e_2^2} = 1$ .

How to quickly identify the conic section represented by

Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0

Compute the discriminant $\Delta = B^2 - 4AC$ . If $\Delta < 0$ , it represents an ellipse (a circle if $A=C$ and $B=0$ ); if $\Delta = 0$ , a parabola; and if $\Delta > 0$ , a hyperbola (provided the general non-degenerate condition $\Delta_{3\times3} \neq 0$ holds).

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