<p>\(r\) be radius of incircle of triangle formed by joining centres of \((x-a)^2+(y-b)^2=9\), \((x-a)^2+(y-b-7)^2=16\) and circle touching above two circles and having radius 5 units. Find \(r^2/4\).</p>
Step-by-Step Solution
Key Concept: The three circles have centers that form a triangle, and we need to find the incircle radius of this triangle. Use the distance formula between centers and the tangency conditions to determine the triangle's side lengths, then apply the incircle radius formula r = Area/s.
<p><strong>Step 1: Identify the three circles.</strong></p><p>Circle 1: Center C₁ = (a, b), radius r₁ = 3</p><p>Circle 2: Center C₂ = (a, b+7), radius r₂ = 4</p><p>Circle 3: Center C₃ = (x₃, y₃), radius r₃ = 5 (to be determined)</p><p><strong>Step 2: Apply tangency conditions.</strong></p><p>Since Circle 3 touches Circle 1 externally: |C₁C₃| = r₁ + r₃ = 3 + 5 = 8</p><p>Since Circle 3 touches Circle 2 externally: |C₂C₃| = r₂ + r₃ = 4 + 5 = 9</p><p><strong>Step 3: Find the triangle sides.</strong></p><p>|C₁C₂| = √[(a-a)² + (b+7-b)²] = 7</p><p>|C₁C₃| = 8</p><p>|C₂C₃| = 9</p><p><strong>Step 4: Calculate the area using Heron's formula.</strong></p><p>Semi-perimeter: s = (7 + 8 + 9)/2 = 12</p><p>Area = √[s(s-a)(s-b)(s-c)] = √[12 × 5 × 4 × 3] = √720 = 12√5</p><p><strong>Step 5: Find the incircle radius.</strong></p><p>r = Area/s = 12√5/12 = √5</p><p><strong>Step 6: Calculate r²/4.</strong></p><p>r²/4 = 5/4</p><p>∴ Answer: <strong>5/4</strong></p>
Correct Answer: 5