Question

The probability that length of a randomly selected chord of a circle lies between $\frac{1}{4}$ and $\frac{3}{4}$ of its diameter is

• A. $\frac{1}{16}$
• B. $\frac{1}{2}$
• C. $\frac{9}{16}$
• D. None of these

Answer 1

Answer: (B)

$\frac{1}{2}$

Let O be the centre of circle

Let Diameter be D

AD be the 3D/4 and BC be D/4

Perpendicular distance of AD from O is ${OM}^2$=$\frac{d^2}{4}-\frac{d^2}{64}$

Perpendicular distance of BC from O is ${ON}^2$=$\frac{d^2}{4}-\frac{{9d}^2}{64}$

Probability that length of a randomly selected chord of a circle lies between 1414 and 3434 of its diameter is area of concentric circle with radius OM and ON respectively by total

P=$\frac{\Pi (O{M}^2-O{N}^2)}{\Pi \ast \frac{d^2}{4}}=\frac{\Pi \left(\right(\frac{d^2}{4}-\frac{d^2}{64})-(\frac{d^2}{4}-\frac{9d^2}{64}\left)\right)}{\Pi \ast \frac{d^2}{4}}=\frac{\Pi \ast \frac{d^2}{8}}{\Pi \ast \frac{d^2}{4}}=\frac{1}{2}$