JEE Main: Circles Touching Axes & Each Other — Smart Geometry Method 💡
❓ Question FOR: Let C 1 C_1 be the circle in the third quadrant of radius 3 , that touches both coordinate axes . Let C 2 C_2 be the circle with centre ( 1 , 3 ) (1,\,3) that touches C 1 C_1 externally at the point ( α , β ) (\alpha,\,\beta) . If ( β − α ) 2 = m n , gcd ( m , n ) = 1 , (\beta - \alpha)^2 = \frac{m}{n}, \quad \gcd(m,n)=1, then the value of m + n m+n is equal to ? 🖼️ Question Image ✍️ Short Solution This problem uses pure coordinate geometry logic : 👉 A circle touching both axes has its centre fixed by symmetry. 👉 When two circles touch externally , the point of contact lies on the line joining their centres . 👉 We find that point using section formula , then compute ( β − α ) 2 (\beta-\alpha)^2 . 🔹 Step 1 — Equation and centre of C 1 C_1 Circle C 1 C_1 C 1 : Lies in third quadrant Touches x-axis and y-axis Radius = 3 Hence, its centre must be: ( − 3 , − 3 ) (-3,\,-3) (Only this point keeps t...