Step 1: Understand the pair of lines. The equation $2x^2-5xy+2y^2=0$ represents two straight lines through the origin. We want the product of distances from $(2,-1)$ to these two lines. Step 2: Factor into two lines. Split the middle term: $2x^2-4xy-xy+2y^2=2x(x-2y)-y(x-2y)=(2x-y)(x-2y)$. So the lines are $2x-y=0$ and $x-2y=0$. Step 3: Distance to the first line. For $2x-y=0$ at $(2,-1)$: $\frac{|2(2)-(-1)|}{\sqrt{2^2+(-1)^2}}=\frac{|5|}{\sqrt{5}}=\frac{5}{\sqrt{5}}=\sqrt{5}$. Step 4: Distance to the second line. For $x-2y=0$ at $(2,-1)$: $\frac{|2-2(-1)|}{\sqrt{1^2+(-2)^2}}=\frac{|4|}{\sqrt{5}}=\frac{4}{\sqrt{5}}$. Step 5: Multiply the two distances. Product $=\sqrt{5}\cdot\frac{4}{\sqrt{5}}=4$. Step 6: State the answer. The product of the perpendicular distances is $4$ units, which is option (C). \[ \boxed{\,4\ \text{units}\,} \]