Two points are randomly chosen on the circumf | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let the two points be $A$ and $B$ on the circumference of a circle with center $O$ and radius $r$.Let $	heta$ be the central angle $\angle AOB$ s

Vedclass · Accepted Answer

$\frac{2}{3}$

Vedclass · Answer

(D) Let the two points be $A$ and $B$ on the circumference of a circle with center $O$ and radius $r$.Let $	heta$ be the central angle $\angle AOB$ subtended by the chord $AB$ at the center.The length of the chord $AB$ is given by $2r \sin(\frac{	heta}{2})$.We want the distance $AB \ge r$,which implies $2r \sin(\frac{	heta}{2}) \ge r$.This simplifies to $\sin(\frac{	heta}{2}) \ge \frac{1}{2}$,which means $\frac{	heta}{2} \ge 30^{\circ}$ or $	heta \ge 60^{\circ}$.Since the points are chosen randomly,the angle $	heta$ can range from $0$ to $180^{\circ}$ (as the distance is symmetric for $	heta$ and $360^{\circ}-	heta$).The total range of the angle $	heta$ is $180^{\circ}$.The favorable range for $	heta$ is $60^{\circ} \le 	heta \le 180^{\circ}$.The probability is the ratio of the favorable range to the total range: $P = \frac{180^{\circ} - 60^{\circ}}{180^{\circ}} = \frac{120^{\circ}}{180^{\circ}} = \frac{2}{3}$.

Two points are randomly chosen on the circumference of a circle of radius $r$. The probability that the distance between the two points is at least $r$ is equal to

Similar Questions

If two sections of strengths $30$ and $45$ are formed from $75$ students who are admitted in a school,then the probability that two particular students are always together in the same section is

In a game of cards,what is the probability that a specific player holds all $4$ kings in a hand of $13$ cards?

From a group of $10$ men and $5$ women,a four-member committee is to be formed such that it must contain at least one woman. The probability that the committee has more women than men is:

An urn contains nine balls of which three are red,four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colours is

Two persons each make a single throw with a die. The probability they get equal value is $p_1$. Four persons each make a single throw and the probability of exactly three being equal is $p_2$,then:

Vedclass Test Series

Exam Paper Generator

Online Exam Module