Infinite Solutions + Geometry Trick in 59 Seconds! 🔥

❓ Concept

🎬 Infinite Solutions + Geometry Trick in 59 Sec

Teen equations, teen variables…
par answer bole “infinitely many solutions”?

👉 Samajh jao — determinant zero ho chuka hai
aur geometry ka scene start ho gaya hai 🔥

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✍️ Short Explanation

Aise questions mein students sirf algebra pe atak jaate hain,
jabki JEE ka smart move hota hai:

👉 Algebra se parameters nikaalo
👉 Geometry se final answer (radius, distance, etc.)

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🔹 Step 1 — Infinite Solutions Condition (CORE RULE 💯)**

System:

$$AX = B$$

Infinite solutions tab milte hain jab:

$$|A| = 0 \quad \text{and} \quad \operatorname{rank}(A)=\operatorname{rank}(A|B)<\text{number of variables}$$

📌 Jaise hi yeh dikhe:

• System dependent hai

• Parameters (λ, μ, …) enter karte hain

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🔹 Step 2 — Geometric Meaning (VERY IMPORTANT 🔥)**

3 variables ⇒ 3D space

Har linear equation ⇒ ek plane

👉 Infinite solutions ka matlab:

• Planes single point pe nahi

• Balki ek LINE ke along intersect kar rahe hain

So:

• Unique solution ❌

• No solution ❌

• Line of solutions ✔️

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🔹 Step 3 — Finding Parameters (λ, μ)**

Jab $|A|=0$:

• Coefficient rows/columns linearly dependent hote hain

• Matlab:

$$\text{One equation} = \text{linear combination of others}$$

👉 Coefficients compare karo
👉 RHS bhi compare karo

Isse:

$$\lambda,\ \mu$$

jaise parameters directly mil jaate hain

📌 Yeh purely algebraic step hai.

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🔹 Step 4 — Link to Coordinate Geometry (SMART TURN 😎)**

Ab twist aata hai 👇

Jo parameters mile:

$$(\lambda,\mu)$$

👉 Yeh 2D plane ka ek point ban jaata hai

Aur question often deta hai:

$$ax+by+c=0$$

(as a line)

Ab problem Matrices se Geometry mein shift ho jaata hai 🔁

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🔹 Step 5 — Radius of Circle Touching a Line

Agar bola ho:

“Circle with centre $(\lambda,\mu)$ touches the line”

Toh:

• Radius = perpendicular distance from centre to line

Formula:

$$r=\frac{|a\lambda+b\mu+c|}{\sqrt{a^2+b^2}}$$

📌 Yeh standard distance-from-point-to-line formula hai —
no derivation needed.

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✅ Final Takeaway

🧠 Infinite Solutions + Geometry Golden Rules

• $|A|=0$ ⇒ system dependent

• Infinite solutions ⇒ planes intersect in a line

• Parameters ⇒ algebra se

• Final quantity ⇒ geometry se

• Radius ⇒ distance formula