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Q.

Five circle C₁, C₂, C₃, C₄, C₅ with radii r₁, r₂, r₃, r₄, r₅ respectively (r₁< r₂ < r₃ < r₄ < r₅) be such that C_i+1 touch each other externally for all i = 1,2,3,4 if all the five circles touch each of the two straight lines L₁ and L₂ and r₁= 2 and r₅ = 32 then r₃ is equal to

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a

8

b

17

c

Depends upon r₂ and r₄

d

$\frac{64}{17}$

answer is A.

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Detailed Solution

Let $C_{1} = x^{2} + y^{2} \equiv 4$
$\begin{array}{l} \Rightarrow C_{2} = {[x - (r_{2} + 2)]}^{2} + (y)^{2} = {(r_{2} + 2)}^{2} \\ \Rightarrow C_{3} = {(x - (2 + 2 r_{2} + r_{3}))}^{2} + y^{2} = {(2 + 2 r_{2} + r_{3})}^{2} \end{array}$
Let y = mx + k be a list to the tangent circle
$\begin{array}{l} \frac{k}{\sqrt{1 + m^{2}}} = \pm 2 \frac{m (2 + r_{2}) + k}{\sqrt{1 + m^{2}}} = \pm r_{2} \\ \Rightarrow 2 r_{1}^{2} = 4 r_{1} \\ \Rightarrow r_{2}^{2} = 2 r_{3} ∴ r_{1}, r_{2}, r_{3} are G.P \end{array}$

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Five circle C1, C2, C3, C4, C5 with radii r1, r2, r3, r4, r5 respectively (r1< r2 < r3 < r4 < r5) be such that Ci+1 touch each other externally for all i = 1,2,3,4 if all the five circles touch each of the two straight lines L1 and L2 and r1= 2 and r5 = 32 then r3 is equal to