JEE Maths: Parametric Trig Forms → Straight Line in 1 Minute! 💡

❓ Concept

⭐ Parametric Trigonometric Curves in Coordinate Geometry

Trig Parametric Curves in 59 Sec

Whenever you see

$$x = f(\theta),\quad y = g(\theta)$$

samajh jao — curve directly nahi diya, balki parameter ke through hide karke diya gaya hai.

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1️⃣ Parametric Form — Actual Meaning

In parametric equations:

• $x$ and $y$ are written in terms of a parameter (usually $\theta$)

• As $\theta$ varies → point $(x,y)$ moves

• The path traced by this moving point is the required curve

👉 Direct graph nahi, motion-based definition hoti hai.

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2️⃣ Why tan(θ + constant) is Used

In JEE, parametric trig curves often look like:

$$x = a\tan(\theta + A),\quad y = b\tan(\theta + B)$$

Reason 👇

• tan-shift identities relate angles linearly

• Difference of angles becomes constant

• This allows easy elimination of $\theta$
  

📌 Especially powerful when:

$$(\theta + A) - (\theta + B) = \text{constant}$$

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3️⃣ Elimination of Parameter (MOST IMPORTANT STEP 🔥)

🎯 Goal: Remove $\theta$, get relation only between $x$ and $y$.

Standard method:

• Let

  $$u = \tan(\theta + A),\quad v = \tan(\theta + B)$$
  

• Use identity:

$$\tan(u - v) = \frac{\tan u - \tan v}{1 + \tan u \tan v}$$

This converts parametric form into:

$$\text{relation between } u \text{ and } v \Rightarrow \text{relation between } x \text{ and } y$$

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4️⃣ Resulting Equation — What Do We Usually Get?

After eliminating $\theta$, you often get:

$$\alpha xy + \beta x + \gamma y + \delta = 0$$

👉 Which represents:

• A straight line

• Or a degenerate conic

📌 This happens because tan-shift identities produce linear + product terms.

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5️⃣ Why This Is Extremely Useful in JEE

🔥 This concept appears in:

• Locus questions

• Parametric coordinate geometry

• Trig + coordinate mixed problems

• Conic transformation questions

Typical JEE pattern:

$$\text{Parametric form} \Rightarrow \text{Eliminate }\theta \Rightarrow \text{Compare coefficients}$$

From there, questions ask:

• Value of constants

• Nature of curve

• Special points on curve

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✅ Final Takeaway

🧠 Master Rule:

• Parametric trig form ≠ scary

• tan-shift → elimination

• Eliminate $\theta$ → get line/conic

• Compare coefficients → answer

Once you crack this, trig + coordinate geometry merge into one topic.