Solution:

The answer is $\mathbf{( C )}$. The relation $x^{2}+y^{2} \leq 100$ represents a circle with center at the origin and radius $10$. The relation $\pi x+\sqrt{17} y \leq 0$ represents a half-plane bounded by a line passing through the origin, which therefore divides the circle into two equal parts. Consequently the required area is half the area of the circle, that is
$$A=\frac{1}{2} \pi 10^{2}=50 \pi$$
Since $3<\pi<4$, we then have $150<A<200$.