Question

For each positive real number ' $k$ ', let $C_k$ denotes the circle with centre at origin and radius '$k$' unit, On a circle $C_k$, a particle moves ' $k$ ' unit in the counterclockwise direction. After completing its motion on $C_k$, the particle moves onto the circle $C_{k+r}$ in some well defined manner, $r>0.$ The motion of the particle continues in this manner.
If $k$ be a positive integer and $r\in R$ and particles moves tangentially from $C_k$ to $C_{k+r}$ such that the length of tangent is equal to ' $k$ ' unit itself. If particle starts from the point $(2,0)$, then :

• A. It will cross the positive $X$-axis again for $2\sqrt{2}<x<4$
• B. It will cross the positive $X$-axis again at $x=3$
• C. It will cross the positive $X$-axis again for $4<x<$ $4\sqrt{2}$
• D. It will cross the positive $X$-axis again at $x=4\sqrt{2}$

Answer 1

Answer: (D)

The particle will travel an angular displacement of $\left(\frac{\pi }{4}+1\right)$ radian while moving from one circle to another. It will cross the positive X-axis again when it travels $2p$ radian angular displacement. We have $\left[\frac{2\pi }{\frac{\pi }{4}+1}\right]=3$, where $[x]$ represents the integral part of $x$. Thus, the particle will be on the fourth circle while crossing the positive $X$-axis again. The radius of the fourth circle will be given by
$r=2(\sqrt{2}{)}^2=4\sqrt{2}$

Hence the particle crosses the positive X-axis at $(4\sqrt{2},0)$