1. Tardigrade
  2. Question
  3. Mathematics
  4. For each positive real number ' k ', let C k denotes the circle with centre at origin and radius 'k' unit, On a circle Ck, 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 ∈ R and particles moves tangentially from Ck to Ck+r such that the length of tangent is equal to ' k ' unit itself. If particle starts from the point (2,0), then :

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Q. 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 :

Conic Sections

Solution:

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