Mathematics - Circle Question with Solution | TestHub
The first unit circle has centre (1,1) and touches both positive coordinate axes. Let P=(0,4). Let x1 be the positive x-intercept of its non-axis tangent from P. For n>=2, the nth unit circle touches the x-axis, the preceding tangent from P, and a new tangent from P crossing the positive x-axis farther right. If xn is the new tangent's x-intercept, find xn in terms of x(n-1).
Options:
Answer:
Solution:
Put u=x(n-1)>0, v=xn>u. The intervening unit circle has centre (h,1). The preceding line is 4x+uy-4u=0. Its centre-distance gives 4h-3u=sqrt(u^2+16). For the next tangent, the opposite side gives 3v-4h=sqrt(v^2+16). Thus 3v-sqrt(v^2+16)=3u+sqrt(u^2+16). Put d=sqrt(u^2+16). v=(5u+3d)/4 works, since sqrt(v^2+16)=(3u+5d)/4. It has v>u. The left side has derivative 3-v/sqrt(v^2+16)>0, so this is unique. Answer A.
