Question

A circle of radius $3$ is placed at the centre of a circle of radius $r>3$ such that the length of a chord of the larger circle tangent to the smaller circle is $8$. The probability that point randomly selected from inside of the larger circle lies inside the annular region of the two circles is

• A. $\frac{3}{4}$
• B. $\frac{9}{25}$
• C. $\frac{16}{25}$
• D. $\frac{7}{3}$

Answer 1

The correct option is C $\frac{16}{25}$

REF. Image

Given radius of the smaller circle = 3

length of chord is 8.

So, $AD=\frac{1}{2}AB=4$

Applying pythogras theorem for ${△}^{le}AOD\Rightarrow AO=5$

So, radius of larger circle = 5

probability that it lies in annular region

$=\frac{Areaofannualarregion}{Areaoflargercircle}$

$=\frac{\pi \left(25\right)-\pi \left(9\right)}{\pi \left(25\right)}=\frac{16}{25}$