<p>Two circles with centres \(C_1\) and \(C_2\) have radii \(r_1\) and \(r_2\) respectively. The circles are such that \(C_1C_2 = r_1 + r_2\) and \(\sqrt{a^2 + b^2} = 2 \pm \sqrt{a^2 + b^2 - 2}\). If \(4r_2 = 2\), find the value of \(4r_2\).</p>
Step-by-Step Solution
Key Concept: When two circles are externally tangent (C₁C₂ = r₁ + r₂), they touch at exactly one point. The given constraint equation with the radical expression determines the relationship between radii through the tangency condition.
<p><strong>Step 1:</strong> Recognize that C₁C₂ = r₁ + r₂ means the two circles are externally tangent (touching at exactly one external point).</p><p><strong>Step 2:</strong> The given equation √(a² + b²) = 2 ± √(a² + b²) - 2 appears to reference the distance relationship. Interpreting this as relating to the tangency condition, this resolves to a specific constraint on the radii.</p><p><strong>Step 3:</strong> From the problem statement, we're given that 4r₂ = 2, which directly yields r₂ = 1/2.</p><p><strong>Step 4:</strong> The question asks us to "find the value of 4r₂," which is already stated as 2 in the problem setup.</p><p>∴ Answer: <strong>B</strong> (4r₂ = 2, or equivalently r₂ = 1/2)</p>
Correct Answer: B