If a circle $$C0$$ with radius $$1$$ unit touches both the axes and as well as line $$L1$$ through $$P(0$$,$$4)$$ which cuts the x $$-axis$$ at $$x1$$,$$0$$ . Again a circle $$C1$$ is drawn touching x $$-axis$$, line $$L1$$ and another line $$L2$$ thorugh point P.$$L2$$ intersects x−axis at $$x2$$,$$0$$ and this process is repeated n times. at $$x2$$,$$0$$ and this process is repeated n times. The centre of circle $$C2$$ is $$31k$$,$$1$$, hence k $$=$$

Question

If a circle $$C0$$ with radius $$1$$ unit touches both the axes and as well as line $$L1$$ through $$P(0$$,$$4)$$ which cuts the x $$-axis$$ at $$x1$$,$$0$$ . Again a circle $$C1$$ is drawn   touching x $$-axis$$, line $$L1$$ and another line $$L2$$ thorugh point P.$$L2$$ intersects x−axis  at $$x2$$,$$0$$ and this process is repeated n times.  at $$x2$$,$$0$$ and this process is repeated n times.  The centre of circle $$C2$$ is $$31k$$,$$1$$, hence k $$=$$

Infinity Learn · Accepted Answer

For Cn circle let centre be an,$$1$$ and it touches line Ln and $$Ln+1$$ i.e. $$xxn+y4=1$$,$$xxn+1+y4=1$$⇒$$4$$α$$n+xn$$−$$4xn16+xn2=1$$⇒$$4$$α$$n+xn+1$$−$$4xn+116+xn+12=1$$ ⇒$$3xn+1$$−$$xn=16+xn2+16+xn+12$$ ⇒$$tan$$⁡θ$$n+12=12tan$$⁡θ$$n2$$ Let $$xn=4cot$$⁡θ$$n0$$<θn<π$$2$$ As $$x1=3$$⇒$$tan$$⁡θ$$12=12$$⇒$$tan$$⁡θ$$n2=12n$$⇒$$xn=22n$$−$$12n$$−$$1x2=152$$ Puttingis (i), we get α$$2=314$$

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