Question

A circle with area A, is contained in the interior of a larger circle with area $A_1+A_2$. If the radius of the larger circle is 3 units and $A_1,A_2,A_1+A_2$, are in A.P., then the radius of the smaller circle is _____.

• A. $\frac{\sqrt{3}}{2}$ units
• B. 1 units
• C. $\frac{2}{\sqrt{3}}$ units
• D. $\sqrt{3}$ units

Answer 1

The correct option is D $\sqrt{3}$ units

$A_1,A_2,A_1+A_2$ are in A.P.

$=>A_2-A_1=A_1+A_2-A_2=>A_2=2A_1$

The smaller circle is $A_1$.

Now, the area of the bigger circle is $A_1+A_2=3A_1=3^2\pi \therefore A_1=\pi \times 3=3\pi$

The area of the smallest circle is $\pi r^2=3\pi =>r=\sqrt{3}units$