How to Solve: Parabola (Parametric Form, Focus, Directrix, Latus Rectum, Tangent, Chord) – IIT JEE Guide

How to Solve: Parabola (Parametric Form, Focus, Directrix, Latus Rectum, Tangent, Chord) – IIT JEE Guide

Introduction

Mastering parabolas in parametric form unlocks 10-15 marks in IIT JEE—enough to push you from a 90th to 99th percentile rank. Whether it’s finding the focus of a disguised parabola or deriving the tangent equation in 30 seconds, this topic separates the top scorers from the rest.

What You Need To Know First

1. Standard equation of a parabola (y² = 4ax, x² = 4ay)
2. Basic coordinate geometry (distance formula, slope of a line)
3. Parametric equations (how to represent curves using a parameter, e.g., t)

Key Vocabulary

Formulas To Know

1. Standard Equations

MEMORISE THIS – These are the base cases for all problems.

2. Parametric Form

For y² = 4ax: - Point on parabola: (at², 2at) - Slope of tangent at (at², 2at): 1/t - Equation of tangent: ty = x + at² - Equation of normal: y = -tx + 2at + at³

For x² = 4ay: - Point on parabola: (2at, at²) - Slope of tangent at (2at, at²): t - Equation of tangent: y = tx - at²

MEMORISE THIS – Parametric forms simplify tangent and chord problems.

3. Chord of Contact

If two tangents are drawn from an external point (x₁, y₁) to the parabola y² = 4ax, the chord joining the points of contact is: Equation: yy₁ = 2a(x + x₁)

MEMORISE THIS – This is frequently asked in JEE.

4. Condition for Tangency

A line y = mx + c is tangent to y² = 4ax if: c = a/m

MEMORISE THIS – This is gold for quick elimination in MCQs.

5. Length of Latus Rectum

For y² = 4ax, length = 4a. For x² = 4ay, length = 4a.

MEMORISE THIS – It’s always 4a for standard parabolas.

Step-by-Step Method

Step 1: Identify the Standard Form

• Check if the equation is y² = 4ax (right-opening) or x² = 4ay (up-opening).
• If not, complete the square or rewrite to match standard form.

Step 2: Find Focus and Directrix

• For y² = 4ax:
• Focus = (a, 0)
• Directrix = x = -a
• For x² = 4ay:
• Focus = (0, a)
• Directrix = y = -a

Step 3: Use Parametric Form for Points

• For y² = 4ax, any point on the parabola is (at², 2at).
• For x² = 4ay, any point is (2at, at²).

Step 4: Find Tangent at a Point

• For y² = 4ax, tangent at (at², 2at) is ty = x + at².
• For x² = 4ay, tangent at (2at, at²) is y = tx - at².

Step 5: Find Chord Joining Two Points

• For y² = 4ax, chord joining (at₁², 2at₁) and (at₂², 2at₂) is: (t₁ + t₂)y = 2x + 2at₁t₂

Step 6: Find Condition for Tangency

• For y² = 4ax, line y = mx + c is tangent if c = a/m.

Step 7: Find Latus Rectum

• Length = 4a (for standard forms).

Worked Examples

Example 1 – Basic (Find Focus, Directrix, Latus Rectum)

Problem: Find the focus, directrix, and latus rectum of y² = 8x.

Solution: 1. Compare with y² = 4ax → 4a = 8 → a = 2. 2. Focus = (a, 0) = (2, 0). 3. Directrix = x = -a → x = -2. 4. Latus rectum = 4a = 8.

What we did and why: - We matched the given equation with the standard form to find a. - Then, we directly applied the formulas for focus, directrix, and latus rectum.

Example 2 – Medium (Tangent in Parametric Form)

Problem: Find the equation of the tangent to y² = 12x at the point (3, 6).

Solution: 1. Compare with y² = 4ax → 4a = 12 → a = 3. 2. Parametric form: (at², 2at) = (3, 6).
- 2at = 6 → 2(3)t = 6 → t = 1. 3. Equation of tangent: ty = x + at² → (1)y = x + 3(1)² → y = x + 3.

What we did and why: - We found the parameter t using the given point. - Then, we plugged t into the tangent formula to get the equation.

Example 3 – Exam-Style (Chord of Contact)

Problem: From the point (4, 6), two tangents are drawn to the parabola y² = 8x. Find the equation of the chord of contact.

Solution: 1. Compare with y² = 4ax → 4a = 8 → a = 2. 2. Chord of contact formula: yy₁ = 2a(x + x₁).
- Here, (x₁, y₁) = (4, 6). 3. Plug in: y(6) = 2(2)(x + 4) → 6y = 4x + 16 → 3y = 2x + 8.

What we did and why: - We recognized that the problem was about the chord of contact. - We applied the formula directly using the external point (4, 6).

Common Mistakes

Exam Traps

1-Minute Recap (Night Before Exam)

"Listen up—this is your 60-second parabola survival guide for JEE.

1. Standard forms: y² = 4ax (right), x² = 4ay (up). Memorise focus, directrix, latus rectum.
2. Parametric form: For y² = 4ax, point is (at², 2at). Tangent: ty = x + at². Chord: (t₁ + t₂)y = 2x + 2at₁t₂.
3. Tangency condition: For y = mx + c, c = a/m if tangent to y² = 4ax.
4. Chord of contact: From (x₁, y₁), equation is yy₁ = 2a(x + x₁).
5. Latus rectum: Always 4a—no exceptions.

If you see a parabola, ask: - Is it standard? If not, rewrite it. - Do I need a tangent? Use parametric form. - Is there an external point? Chord of contact formula.

You’ve got this—go crush it!