<p>In a triangle, the sum of lengths of two sides is <i>x</i> and the product of the lengths of the same two sides is <i>y</i>. If <i>x</i>² - <i>c</i>² = <i>y</i>, where <i>c</i> is the length of the third side of the triangle, then the circumradius of the triangle is (JEE Main 2019)</p>
Step-by-Step Solution
Key Concept: Use the Law of Cosines to relate the three sides and angles of the triangle, then apply the Law of Sines to find the circumradius. The given condition x² - c² = y constrains the angle opposite to side c.
<p><strong>Step 1:</strong> Let the two sides be a and b, with sum a + b = x and product ab = y. The third side is c.</p><p><strong>Step 2:</strong> Expand x²: x² = (a + b)² = a² + 2ab + b² = (a² + b²) + 2y</p><p><strong>Step 3:</strong> From the given condition x² - c² = y, substitute: (a² + b²) + 2y - c² = y, which simplifies to a² + b² + y = c²</p><p><strong>Step 4:</strong> Apply the Law of Cosines: c² = a² + b² - 2ab·cos(C), where C is the angle opposite side c.</p><p><strong>Step 5:</strong> Substitute from Step 3: a² + b² + y = a² + b² - 2ab·cos(C)</p><p><strong>Step 6:</strong> This gives y = -2ab·cos(C), so y = -2y·cos(C), which means cos(C) = -1/2</p><p><strong>Step 7:</strong> Therefore C = 120° (or 2π/3 radians)</p><p><strong>Step 8:</strong> Apply the Law of Sines: c/sin(C) = 2R, where R is the circumradius</p><p><strong>Step 9:</strong> Since sin(120°) = √3/2, we have: 2R = c/(√3/2) = 2c/√3</p><p><strong>Step 10:</strong> Therefore R = c/√3 = c√3/3</p><p><strong>∴ Answer:</strong> B</p>
Correct Answer: B