You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard X

Mathematics

Touching Circles Theorem

For each natu...

Question

For each natural number k, let $C_{k}$ denote a circle with radius k centimeters and center at origin O. On the circle $C_{k}$ a partical moves k centimeters in the counterclockwise direction. After completing its motion on $C_{k}$ , the particle moves to $C_{k + 1}$ in the radial direction. The motion of the particle continues in this manner. The particle starts at (1, 0). If the particle crosses the positive direction of the x-axis for the first time on the circle $C_{n}$ , then find the value of n.

Open in App

Solution

The motion of the particle on the first four circles is shown with boil lines in Fig. Note that on every circle the particle travels just 1 rad. The particle crosses the positive direction of the x-axis first time on $C_{n}$ , where n is the least positive integer such that $n \geq 2 π \Rightarrow n = 7$ .
Ans: 7

Suggest Corrections

Similar questions

Q. For each natural number

k

,let

C_{k}

denote the circle with radius

k

centimeters and centre at the origin

O

. on the circle

C_{k}

a particle moves

k

centimetres in the counter-clockwise direction. After completing its motion on

C_{k}

, the particle moves to

C_{k + 1}

in the radial direction (away from centre). The motion of the particle continues in this manner. The particle starts at

(1, 0)

. If the particle crosses the positive direction of x-axis for the first time on the circle

C_{n}

, then

n =

Q. For each natural number

k

, let

C_{k}

denote the circle with radius

k c m

, and centre at the origin. On the circle

C_{k}

a particle moves

k c m

, in the center-clockwise direction. After completing its motion on

C_{k}

, the particle moves to

C_{k + 1}

in the radial direction. The motion of the particle continuous in this manner. The particle starts at

(1, 0)

. If the particle crosses the positive direction of the

x -

axis for the first time on the circle

C_{k}

, then find

n

Q. For each

k ϵ N

, let

C_{k}

denote the circle whose equation is

x^{2} + y^{2} = k^{2}

. On the circle

C_{k}

, a particle moves

k

units in the anticlockwise direction. After completing its motion on

C_{k}

, the particle moves to

C_{k + 1}

in the radial direction. The motion of the particle continues in this manner. The particle starts at

(1, 0)

. If the particle crossed the positive direction of the

x

-axis for the first time of the circle

C_{n}

then

n

is:

Q. If

k

belongs to set of natural numbers and

l = 1

the particle moves in the radial direction from circle

C_{k}

C_{k + 1}

. If particle starts from the point

(- 1, 0)

, then

Q. If

k

belongs to set of natural numbers and

l = R

particle moves tangentially from the circle

C_{k}

C_{k + 1}

such that the length of tangent is equal to

k

units itself.If particle starts from the point

(1, 0)

then

Join BYJU'S Learning Program

The motion of the particle on the first four circles is shown with boil lines in Fig. Note that on every circle the particle travels just 1 rad. The particle crosses the positive direction of the x-axis first time on Cn, where n is the least positive integer such that n≥2π⇒n=7.Ans: 7