Tangent Circles Using Centres and Radii

 

❓ Concept

Do circles sirf ek point par touch karte hain?
👉 Matlab pure geometry ka magic — no heavy algebra needed!

Bas centres + radii + section formula —
aur answer seedha mil jaata hai.

---

🖼️ Concept Image

---

✍️ Short Explanation

External tangency ka matlab:
✔ Circles overlap nahi karte
✔ Bas ek hi point par touch karte hain
✔ Aur centres–contact point ek straight line par hote hain

Isliye equations likhne ki zarurat hi nahi —
pure geometry se crack ho jaata hai.

---

🔹 Step 1 — Circle Touching Coordinate Axes

Agar koi circle x-axis aur y-axis dono ko touch karta hai
toh radius = axis se distance

Isliye centre hota hai:

$$(\pm r,\ \pm r)$$

Sign quadrant par depend karta hai.

Example:
Third quadrant → $(-r,-r)$

---

🔹 Step 2 — External Tangency Condition (MOST IMPORTANT 🔥)**

Do circles externally touch tabhi:

$$\text{Distance between centres} = r_1+r_2$$

Bas — yahi sabse powerful condition hai.

---

🔹 Step 3 — Line of Centres Concept

Centres $C_1$ & $C_2$
aur point of contact $P$

👉 ek hi straight line par hote hain

Aur $P$ line segment $C_1C_2$ ko radii ke ratio mein divide karta hai:

$$C_1P : C_2P = r_1 : r_2$$

---

🔹 Step 4 — Coordinates of Point of Contact (Shortcut 😎)**

Use section formula:

$$P = \frac{r_2 C_1 + r_1 C_2}{r_1+r_2}$$

Isse direct α, β mil jaate hain —
circle equation likhne ki bilkul zaroorat nahi!

---

🔹 Step 5 — JEE Strategy (Guaranteed Score 💯)**

Har touching-circles problem mein:

1️⃣ Pehle centres fix karo
(axes touch → centre = $(\pm r,\pm r)$)

2️⃣ Distance between centres = $r_1+r_2$ lagao
→ radii/coordinates mil jaate hain

3️⃣ Section formula use karke
point of contact nikaalo

4️⃣ Jo expression poocha hai
जैसे $(\beta-\alpha)^2$
→ directly evaluate karo

No messy algebra
👉 Pure logical geometry

---

✅ Final Takeaway

🧠 Golden Rules

• Axis touch → centre = (±r, ±r)

• External tangency → centre distance = $r_1+r_2$

• Centres + contact point = collinear

• Contact point divides line in ratio $r_1:r_2$

• Section formula → direct coordinates

Isse circles ke tough JEE questions bhi lightning-fast ho jaate hain 🚀