Let n geq 3 and let C_1, C_2, ldots, C_n, be | vedclass

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Question

(d)

We have,

$C_1, C_2, C_3, \ldots, C_n$ be circle with radii

$r_1, r_2, \ldots, r_n$ respectively. $C_i$ and $C_{i+1}$ touch externally $X$

Vedclass · Accepted Answer

a geometric progression with common ratio $2+\sqrt{3}$

Vedclass · Answer

(d)

We have,

$C_1, C_2, C_3, \ldots, C_n$ be circle with radii

$r_1, r_2, \ldots, r_n$ respectively. $C_i$ and $C_{i+1}$ touch externally $X$-axis and $y=2 \sqrt{2} x+10$ are tangent of each circle.

Slope of line $y=2 \sqrt{2} x+10$ is $2 \sqrt{2}$

$	herefore \quad 	an 2 	heta =2 \sqrt{2}$

$\frac{2 	an 	heta}{1-	an ^2 	heta} =2 \sqrt{2}$

$\sqrt{2} 	an ^2 	heta+	an 	heta-\sqrt{2}=0$

$(\sqrt{2} 	an 	heta-1)(	an 	heta+\sqrt{2}) =0$

$	an 	heta=\frac{1}{\sqrt{2}} 	an 	heta 
eq-\sqrt{2}$

$\sin 	heta=\frac{1}{\sqrt{3}}$

$\operatorname{In} \Delta P Q M, \sin 	heta=\frac{Q M}{P Q}$

$\Rightarrow \quad P Q=\sqrt{3} Q M \Rightarrow P Q=\sqrt{3} r_1$

$\operatorname{In} 	riangle P R N$,

$\sin 	heta=\frac{R N}{P R}=\frac{r_2}{P Q+r_1+r_2}=\frac{r_2}{\sqrt{3} r_1+r_1+r_2}$

$\Rightarrow \quad \frac{1}{\sqrt{3}}=\frac{r_2}{(\sqrt{3}+1) r_1+r_2}$

$\Rightarrow r_2+(\sqrt{3}+1) r_1=\sqrt{3}-r_2$

$\Rightarrow \quad \frac{r_2}{r_1}=\frac{\sqrt{3}+1}{\sqrt{3}-1}$

$r_1, r_2, r_3$ a geometric progression with common ratio $2+\sqrt{3}$.

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