Purely Real vs Purely Imaginary — One Identity to Crack All Questions! 🔥

 

❓ Concept

🎬 Purely Real / Imaginary Trick

Complex numbers ke questions JEE mein scary lagte hain —
lekin purely real / purely imaginary ka poora game
👉 sirf ek condition + ek rationalizing trick pe based hota hai.

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🖼️ Concept Image

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✍️ Short Explanation

Jab question bole:

• expression purely real hai

• expression purely imaginary hai

Toh yaad rakho 👇
👉 Real part aur Imaginary part ka zero hona hi sab kuch decide karta hai.

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🔹 Step 1 — Basic Form of a Complex Number

Har complex number likha ja sakta hai:

$$z = x + iy$$

Where:

• Real part = $x = \Re(z)$
  

• Imaginary part = $y = \Im(z)$
  

Ye breakdown har question ka base hai.

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🔹 Step 2 — Condition for Purely Real

A complex number purely real tab hota hai jab:

$$\Im(w) = 0$$

👉 Matlab imaginary part bilkul zero.

Examples:

• $5$ ✔️ purely real

• $3 - 0i$ ✔️ purely real

• $2 + i$❌ not purely real

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🔹 Step 3 — Condition for Purely Imaginary

A complex number purely imaginary tab hota hai jab:

$$\Re(w) = 0$$

👉 Matlab real part bilkul zero.

Examples:

• $4i$ ✔️ purely imaginary

• $-7i$✔️ purely imaginary

• $1 + i$ ❌ not purely imaginary

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🔹 Step 4 — Rationalizing Trick (MOST IMPORTANT 🔥)

JEE mein expression aksar aisa hota hai:

$$\frac{z - a}{z + b}$$

Direct real/imaginary part nikaalna mushkil hota hai.
Solution 👇

👉 Numerator aur denominator ko conjugate se multiply karo:

$$\frac{z - a}{z + b} = \frac{(z-a)(\bar z + \bar b)}{|z + b|^2}$$

📌 Ab:

• Denominator purely real ban gaya

• Numerator expand karne pe:

  • Real part alag

  • Imaginary part alag

Phir:

• Imaginary part = 0 ⇒ purely real

• Real part = 0 ⇒ purely imaginary

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🔹 Step 5 — Key JEE Observation (Hidden Geometry 💡)

Bahut powerful observation:

• $|z| = \text{constant}$ ⇒ circle

• Expression purely real ⇒

  • Usually chord(s) of the circle

• Expression purely imaginary ⇒

  • Often diameter-type condition

Isliye:

• Kabhi sirf 2 points

• Kabhi infinitely many points

👉 Final answer geometry decide karti hai, algebra sirf tool hai.

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✅ Final Takeaway

🧠 Golden Rules for JEE:

• Purely real ⇒ imaginary part zero

• Purely imaginary ⇒ real part zero

• Fraction ho ⇒ rationalize

• $|z|=\text{constant}$ ⇒ circle picture banao