<p>The equation of the given line is <em>3x + 4y − 24 = 0</em>. The incentre of the triangle formed by this line with the coordinate axes is:</p>
Step-by-Step Solution
Key Concept: The triangle is formed by a line and the coordinate axes, creating a right triangle at the origin. The incentre of a right triangle with legs a and b has coordinates (r, r) where r is the inradius = (a + b - c)/2, and c is the hypotenuse.
<p><strong>Step 1:</strong> Find the intercepts of line 3x + 4y − 24 = 0 with the coordinate axes.</p><p>When y = 0: 3x = 24 ⟹ x = 8, so A = (8, 0)</p><p>When x = 0: 4y = 24 ⟹ y = 6, so B = (0, 6)</p><p>The third vertex is O = (0, 0) (origin).</p><p><strong>Step 2:</strong> Calculate the sides of triangle OAB.</p><p>OA = 8 (along x-axis), OB = 6 (along y-axis), AB = √(8² + 6²) = √(64 + 36) = √100 = 10</p><p><strong>Step 3:</strong> Find the inradius using r = (a + b − c)/2 where a, b are legs and c is hypotenuse.</p><p>r = (8 + 6 − 10)/2 = 4/2 = 2</p><p><strong>Step 4:</strong> For a right triangle with right angle at origin and legs along the axes, the incentre is at (r, r).</p><p>∴ Incentre = (2, 2) — Answer: B</p>
Correct Answer: B