My approach to this question is the sum of the circumferences of the two smaller circles is $2\pi(a+b)$, where $a$ is the radius of the circle on the left and $b$ is the radius of the one on the right. And we now compare $2\pi(a+b)$ with $2\pi r$, where $r$ is the radius of the largest circle.
However, the answer is C, which means $a+b=r$. I am confused that how do you know $R$ is the center of the largest circle since the question only tells you that the centers lie on line $PQ$.
$R$ is not necessarily the center of the $PQ$ segment. If the radius of the smaller circle is $a$, and the larger circle is $b$, the center of the outside circle is at $a+b$ from either $P$ or $Q$, while point $R$ is at $2b$ from one side and $2a$ from the other side
Note that circumference of a circle $= \pi\times d$ where $d$ is diameter of circle.
One smaller circle has diameter PR and other is QR.
Diameter of the largest circle is PQ. Now PQR lie on the same line.
$PR + QR = PQ$
Now multiply with $\pi$ on both sides.
$\pi\times PR+\pi\times QR=\pi\times PQ$
Circumference of smaller circle with diameter $PR +$ Circumference of smaller circle with diameter $QR =$ Circumference of larger circle with diameter PQ.
Say the big circle has diameter $d$, and the two smaller circles have diameters $a$ and $b$, respectively. So clearly $a + b = d$.
Circumference of big circle: $\pi d$.
Sum of smaller circumferences: $\pi a + \pi b = \pi ( a + b) = \pi d$.
So the two quantities are the same.
assume that the smaller inner circle and the bigger inner circle has the radius $a$ & $b$ the diameter of the outer circle is equal to the sum of the diameter of the inner circles.
thus we can say that $2R=2a+2b$ thus $R=a+b$
now the sum of the circumference of inner circles is $=2\pi a +2\pi b=2\pi (a+b)$
and circumference of the outer circle is $=2\pi R= 2\pi (a+b) $
thus both quantities are equal.
