Circles
Circle
star_batch_jee_advanced_2025
Grade 11
Question:
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: The locus of circle centers that satisfy tangency conditions with a fixed line through the origin determines a straight line whose equation is found from the coordinates of known centers.
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 $x - 2y = 0$.
Correct Answer: 5