Circles
Circumcircle of triangle
Grade 11
Question:
<p>If the incentre of an equilateral triangle is (1, 1) and the equation of its one side is \(3x + 4y + 3 = 0\), then the equation of the circumcircle of this triangle is</p>
<p>\(x^2 + y^2 - 2x - 2y - 2 = 0\)</p>
<p>\(x^2 + y^2 - 2x - 2y - 14 = 0\)</p>
<p>\(x^2 + y^2 - 2x - 2y + 2 = 0\)</p>
<p>\(x^2 + y^2 - 2x - 2y - 7 = 0\)</p>
Step-by-Step Solution
Key Concept: For an equilateral triangle, the incentre and circumcentre coincide. Use the perpendicular distance from the incentre to a side as the inradius, then find the circumradius using the relationship R = 2r for equilateral triangles.
<p><strong>Step 1:</strong> Find the inradius by calculating perpendicular distance from incentre I(1,1) to the line 3x + 4y + 3 = 0.</p><p>Using distance formula: r = |3(1) + 4(1) + 3|/√(9 + 16) = |3 + 4 + 3|/5 = 10/5 = 2</p><p><strong>Step 2:</strong> For an equilateral triangle, the circumradius R = 2r (incentre = circumcentre).</p><p>Therefore, R = 2 × 2 = 4</p><p><strong>Step 3:</strong> The circumcircle has centre at (1, 1) and radius 4.</p><p>Equation: (x - 1)² + (y - 1)² = 16</p><p>Expanding: x² + y² - 2x - 2y + 1 + 1 - 16 = 0</p><p>∴ x² + y² - 2x - 2y - 14 = 0</p>
Correct Answer: B