As shown in the figure, three circles which have the same radius $r$ have centres at $(0,0)$, $(1,1)$, and $(2,1)$. If they have a common tangent line, as shown, then the value of $10\sqrt{5r}$ is ___.
Step-by-Step Solution
Key Concept: For three circles of equal radius r with centers at (0,0), (1,1), and (2,1) to share a common tangent line, the perpendicular distance from each center to the tangent line must equal r. This requires solving the distance formula d = |ax₀ + by₀ + c|/√(a² + b²) = r simultaneously for all three centers to find the tangent line and radius.
Three circles with equal radii are arranged such that each successive circle's center lies at a fixed distance along a line. The line joining the origin to the center of circle $C_2$ at $(2,1)$ has slope $\frac{1}{2}$, giving the equation $y = \frac{x}{2}$ or equivalently $x - 2y = 0$.
<div class="key-concept"><strong>Key Concept:</strong> For three circles of equal radius r with centers at (0,0), (1,1), and (2,1) to share a common tangent line, the perpendicular distance from each center to the tangent line must equal r. This requires solving the distance formula d = |ax₀ + by₀ + c|/√(a² + b²) = r simultaneously for all three centers to find the tangent line and radius.</div>
<div class="trap-box"><strong>Trap:</strong> A common error is assuming the tangent line passes through or near the center points, or incorrectly setting up the distance equations without ensuring all three perpendicular distances equal r simultaneously. Students may also confuse the geometry by treating this as a collinearity problem rather than a distance constraint problem, leading to incorrect simplification of the radical expression √(5r).</div>
Correct Answer: 5