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Equation of Family of Circles Touching a Line and Passing through a Given Point on the Line

Five circles ...

Question

Five circles $C_{1}, C_{2}, C_{3}, C_{4}, C_{5}$ with radii $r_{1}, r_{2}, r_{3}, r_{4}, r_{5}$ respectively $(r_{1} < r_{2} < r_{3} < r_{4} < r_{5})$ be such that $C_{i}$ and $C_{i} + 1$ touch each other externally for all $i = 1, 2, 3, 4$ . If all the five circles touches two straight lines $L_{1}$ and $L_{2}$ and $r_{1} = 2$ and $r_{5} = 32$ , then $r_{3}$ is(units)

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Solution

Let the $∠ O_{1} A T_{1} = x$ , then
$sin x = \frac{r_{2} - r_{1}}{r_{1} + r_{2}}$
Similarly,
$sin x = \frac{r_{3} - r_{2}}{r_{3} + r_{2}}$
Now,
$\frac{r_{2} - r_{1}}{r_{1} + r_{2}} = \frac{r_{3} - r_{2}}{r_{3} + r_{2}} \Rightarrow r_{2} r_{3} + r_{2}^{2} - r_{1} r_{3} - r_{1} r_{2} = r_{1} r_{3} - r_{1} r_{2} + r_{2} r_{3} - r_{2}^{2} \Rightarrow r_{2}^{2} = r_{1} r_{3}$
So, $r_{1}, r_{2}, r_{3}$ are in G.P.
Similarly, $r_{1}, r_{2}, r_{3}, r_{4}, r_{5}$ are in G.P.
Therefore,
$r_{3}^{2} = r_{1} r_{5} \Rightarrow r_{3} = 8 units$

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