Parabola Class 11 Notes & Numericals for IIT JEE

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Why Parabola Is a High-Priority Chapter for IIT JEE 

Parabola is part of Conic Sections (Class 11, Chapter 11) — one of the most consistently tested chapters in both JEE Main and JEE Advanced. Together with Ellipse and Hyperbola, Conics contributes 2–4 questions every year in JEE Main and at least 1–2 high-difficulty questions in JEE Advanced.

Among the three non-circular conics, the Parabola is the foundation chapter — mastered first because its equation is the simplest and its properties (tangent, normal, focal chord) carry over structurally to Ellipse and Hyperbola. A student who deeply understands Parabola completes Ellipse and Hyperbola in significantly less time.

For JEE Main, Parabola questions typically test: standard equation identification, tangent and normal equations (all three forms), and properties of focal chords. For JEE Advanced, questions test deeper results — combined locus problems, tangent-normal intersections, and multi-concept problems where Parabola interacts with other chapters.

💡 Expert Tip by eSaral Mathematics Faculty, IIT Kota: "Parabola in JEE is not about memorising 50 results — it is about deeply knowing 8–10 core results and being able to derive everything else from them. The students who score full marks from Conics in JEE are not the ones who memorised the most formulas; they are the ones who understood the parametric form (at², 2at) so deeply that they could derive tangent, normal, and chord equations in the exam hall without looking at a formula sheet."

Parabola Class 11 Notes & Numericals

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India's Best Exam Preparation for Class 11th - Download Now

India's Best Exam Preparation for Class 11th - Download Now

India's Best Exam Preparation for Class 11th - Download Now

India's Best Exam Preparation for Class 11th - Download Now

Basic Definitions and Key Terminology 

Core Definitions

Parabola: The locus of a point P that moves such that its distance from a fixed point (the focus) is always equal to its distance from a fixed line (the directrix).

$$\text{PS}=\text{PM}$$

where S = focus and M = foot of perpendicular from P to the directrix.

Eccentricity: For a parabola, eccentricity e = 1 always. This is the defining property that distinguishes a parabola from an ellipse (e < 1) and a hyperbola (e > 1).

Key Elements of a Parabola

Standard Forms of Parabola — All Four Orientations 

Complete Reference Table

The Most Important Form — y² = 4ax

For the standard parabola y² = 4ax:

• Any point on the parabola can be written as (at², 2at) — the parametric form
• Focal distance of point (x₁, y₁): PS = x₁ + a
• Equation of latus rectum: x = a
• Endpoints of latus rectum: (a, 2a) and (a, −2a)
• Length of latus rectum: 4a

General Second-Degree Conic and Parabola Condition

A second-degree equation ax² + bxy + cy² + dx + ey + f = 0 represents a parabola when:

b2−4ac=0 (discriminant condition)b^2 - 4ac = 0 \text{ (discriminant condition)}b2−4ac=0 (discriminant condition)

Parametric Equations of a Parabola {#parametric}

For y² = 4ax, the parametric form is:

x=at2,y=2atx = at^2, \quad y = 2atx=at2,y=2at

where t is the parameter. Any point on the parabola is written as P(t) = (at², 2at).

Why the Parametric Form Is Essential for JEE

The parametric form simplifies every derivation in this chapter:

• Instead of working with (x, y) coordinates separately, one parameter t describes any point
• Tangent at P(t): ty = x + at²
• Normal at P(t): y + tx = 2at + at³
• Chord joining P(t₁) and P(t₂): (t₁ + t₂)y = 2x + 2at₁t₂

Key Parametric Results

Position of a Point Relative to a Parabola

For the parabola y² = 4ax, a point P(h, k) lies:

Physical interpretation: Inside the parabola means closer to the axis than the parabola curve; outside means farther from the axis.

Tangent to a Parabola — All Three Forms 

For the parabola y² = 4ax:

Form 1 — Point Form (Tangent at a given point on the parabola)

Tangent at point (x₁, y₁) on y² = 4ax:

yy1=2a(x+x1)yy_1 = 2a(x + x_1)yy1​=2a(x+x1​)

Form 2 — Slope Form (Tangent with a given slope m)

y=mx+amy = mx + \frac{a}{m}y=mx+ma​

This line is tangent to y² = 4ax for any value of m ≠ 0.

Point of tangency: (am2,2am)\left(\frac{a}{m^2}, \frac{2a}{m}\right) (m2a​,m2a​)

Condition of tangency: For line y = mx + c to be tangent to y² = 4ax:

c=amc = \frac{a}{m}c=ma​

Form 3 — Parametric Form (Tangent at parameter t)

ty=x+at2ty = x + at^2ty=x+at2

Key Tangent Properties

• Two tangents drawn from an external point to y² = 4ax meet the axis at angles supplementary to the angle between the chord of contact and the axis
• Tangents at the ends of a focal chord of y² = 4ax are perpendicular to each other and meet on the directrix
• Tangent at the vertex (origin) is the y-axis (x = 0)

Normal to a Parabola {#normal}

Equation of Normal — Three Forms

For y² = 4ax:

At point (x₁, y₁):

y−y1=−y12a(x−x1)y - y_1 = -\frac{y_1}{2a}(x - x_1)y−y1​=−2ay1​​(x−x1​)

In slope form (slope = m):

y=mx−2am−am3y = mx - 2am - am^3y=mx−2am−am3

The foot of this normal has coordinates (am², −2am).

In parametric form (at parameter t):

y+tx=2at+at3y + tx = 2at + at^3y+tx=2at+at3

Important Normal Properties

• The normal at point P(t) meets the parabola again at point Q(t') where:

t′=−t−2tt' = -t - \frac{2}{t}t′=−t−t2​

• Three normals can be drawn from an external point to y² = 4ax in general
• If three normals from point (h, k) have slopes m₁, m₂, m₃, then:
  • m₁ + m₂ + m₃ = 0
  • m₁m₂ + m₂m₃ + m₁m₃ = (2a − h)/a
  • m₁m₂m₃ = k/a

Chord of Contact and Chord with Given Midpoint 

Chord of Contact

If tangents are drawn from an external point P(h, k) to y² = 4ax, the chord of contact (line joining the two points of tangency) has equation:

$$ky=2a(x+h)$$

This can be written as T = 0 where T = yy₁ − 2a(x + x₁) evaluated at (h, k).

Chord with Given Midpoint

If the midpoint of a chord of y² = 4ax is M(h, k), the equation of that chord is:

$$k(y-k)=2a(x-h)$$

Or equivalently: T = S₁

where T = yy₁ − 2a(x + x₁) and S₁ = y₁² − 4ax₁ evaluated at (h, k).

Focal Chord Properties

For a focal chord joining points P(t₁) and Q(t₂) on y² = 4ax:

t1⋅t2=−1t_1 \cdot t_2 = -1t1​⋅t2​=−1

This is one of the most important results for JEE — if t₁t₂ = −1, the chord is focal (passes through the focus).

Length of focal chord through parameter t:

ℓ=a(t+1t)2\ell = a\left(t + \frac{1}{t}\right)^2ℓ=a(t+t1​)2

Harmonic mean of focal distances: If SP and SQ are the focal distances of endpoints of a focal chord, then:

1SP+1SQ=1a\frac{1}{SP} + \frac{1}{SQ} = \frac{1}{a}SP1​+SQ1​=a1​

Key Properties and Important Results 

Must-Know Results for JEE — Quick Reference

The Reflection Property — JEE Conceptual Question

The most physically significant property of a parabola: any ray parallel to the axis of a parabola, on reflection from the parabolic surface, passes through the focus. This is why parabolic reflectors are used in satellite dishes, car headlights (reversed: light from focus → parallel rays), and telescopes.

JEE Main and JEE Advanced Weightage 

Most Repeated JEE Main Question Types from Parabola

How to Study Parabola for Maximum JEE Marks 

Step-by-Step Study Plan

Step 1 — Master the standard form y² = 4ax completely (Day 1) Before touching any other form, make y² = 4ax your reference form. Write from memory: focus, directrix, vertex, axis, latus rectum endpoints, focal distance formula, and the parametric point (at², 2at). Everything in this chapter flows from this one form.

Step 2 — Learn all four orientations with one diagram each (Day 1) Draw the four standard parabolas (y² = 4ax, y² = −4ax, x² = 4ay, x² = −4ay) with their focus and directrix labelled. The table in this article covers all four — reproduce it from memory after reading it once.

Step 3 — Derive tangent and normal in all three forms (Day 2) Do not just memorise the tangent formulas — derive them for y² = 4ax:

• Point form by implicit differentiation
• Parametric form by substituting (at², 2at)
• Slope form by using the condition c = a/m

Deriving these yourself locks them in more firmly than any amount of memorisation.

Step 4 — Practise the five most important JEE result types (Day 3) Work through these five result types with 2–3 examples each:

• Tangent in slope form and condition of tangency
• Normal rejoining the parabola (t' = −t − 2/t)
• Focal chord (t₁t₂ = −1)
• Chord of contact from external point
• Locus problems (intersection of two tangents/normals)

Step 5 — Solve JEE Main PYQs on Parabola year-wise (Day 4–5) Access all JEE Main previous year question papers on eSaral and filter for Parabola. Solve year by year from the most recent. After 4–5 years of PYQs, every important question template will feel familiar.

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