Le n≥3 and let C1,C2,....Cn, be circles with radii r1,r2,.....rn, respectively. Assume that Ci and Ci+1 touch externally for 1≤i≤n−1. It is also given that the x-axis and the line y=2√2x+10 are tangential to each of the circles. Then r1,r2,.....,rn are in.
Let us solve the problem with n=2 and we will get a generalised result
Let the angle between y=2√2x+10 and x axis be 2θ
Angle between a line and x axis that is tanθ is equal to the slope of the given line
⇒tan2θ=2√2⇒cos2θ=131−2sin2θ=13⇒sinθ=1√3
From the figure
sinθ=r1AP1=r2AP2=1√3.....(i)AP2=AP1+P1P2AP2=AP1+r1+r2√3r2=√3r1+r1+r2r2(√3−1)=r1(√3+1)r2r1=(√3+1)(√3−1)=(√3+1)(√3−1)×(√3+1)(√3+1)=2+√3
Clealry the radius are in geometric progression and the result will also be followed for the proceeding circles.
Hence, option D is correct