Circles
Chord of a Circle
Grade 11
Question:
<p>The equation of the tangent to the circle \(x^2 + y^2 - 2x - 1 = 0\) at the point (2, 1) is \(x + y - 3 = 0\). This line is a chord of another circle \(C_2\) with centre (3, –2) and \(AB = 4\). Find the radius of circle \(C_2\).</p>
<p>\(\sqrt{3}\)</p>
<p>\(\sqrt{5}\)</p>
<p>\(\sqrt{6}\)</p>
<p>\(\sqrt{7}\)</p>
Step-by-Step Solution
Key Concept: Use the chord-distance formula: if a chord of length AB lies on a line at perpendicular distance d from the center, then r² = d² + (AB/2)². The perpendicular distance from center (3, -2) to line x + y - 3 = 0 is the key.
<p><strong>Step 1:</strong> Find the perpendicular distance from center C₂(3, -2) to the line x + y - 3 = 0.</p><p>Using the point-to-line distance formula: d = |ax₀ + by₀ + c|/√(a² + b²)</p><p>d = |3 + (-2) - 3|/√(1² + 1²) = |-2|/√2 = 2/√2 = √2</p><p><strong>Step 2:</strong> Apply the chord-distance relationship.</p><p>For a chord of length AB = 4 at perpendicular distance d from the center of a circle with radius r:</p><p>r² = d² + (AB/2)²</p><p><strong>Step 3:</strong> Substitute values.</p><p>r² = (√2)² + (4/2)²</p><p>r² = 2 + 4 = 6</p><p>r = √6</p><p>∴ Answer: C (radius = √6)</p>
Correct Answer: C