Consider the circle as shown in the figure and choose CORRECT option for this case.

Let the radius of small circle = $$r_1$$

Radius of larger circle = $$r_2$$

From the figure,

OA > AC

$$\Rightarrow$$  $$r_2$$ > 2$$r_1$$

$$\Rightarrow$$  $$\frac{r_1}{r_2}<\frac{1}{2}$$

From the figure,

QC$$\bot\ $$AP and PB$$\bot\ $$AP

Both QC and PB are perpendicular to same line

$$\Rightarrow$$  QC is parallel PB

$$\triangle$$AQC is similar to $$\triangle$$APB

$$\Rightarrow$$  $$\frac{AC}{AB}=\frac{QC}{PB}$$

$$\Rightarrow$$  $$\frac{2r_1}{2r_2}=\frac{QC}{PB}$$

$$\Rightarrow$$  $$\frac{QC}{PB}=\frac{r_1}{r_2}$$

$$\Rightarrow$$  $$\frac{QC}{PB}<\frac{1}{2}$$

$$\Rightarrow$$  $$QC<\frac{1}{2}PB$$

Hence, the correct answer is Option A