Point A is the center of a circle. If the ratio of the lengths of $\left|\overrightarrow{\mathrm{OA}}\right|$ to $\left|\overrightarrow{\mathrm{OB}}\right|$ is $2:1$, what is the ratio of the areas of the circles with radii $\left|\overrightarrow{\mathrm{OA}}\right|$ and $\left|\overrightarrow{\mathrm{OB}}\right|$?

• A. $1:2$
• B. $2:1$
• C. $4:1$
• D. $1:4$

Answer: (C)

Verified step by step guidance

1. Identify the given ratio of the lengths of the radii: \(|\overrightarrow{OA}| : |\overrightarrow{OB}| = 2 : 1\).
2. Recall that the area \(A\) of a circle is related to its radius \(r\) by the formula \(A = \pi r^{2}\).
3. Express the areas of the two circles in terms of their radii: \(A_{OA} = \pi (|\overrightarrow{OA}|)^{2}\) and \(A_{OB} = \pi (|\overrightarrow{OB}|)^{2}\).
4. Substitute the ratio of the radii into the area expressions to find the ratio of the areas: \(A_{OA} : A_{OB} = (|\overrightarrow{OA}|)^{2} : (|\overrightarrow{OB}|)^{2} = (2)^{2} : (1)^{2}\).
5. Simplify the ratio of the areas to get the final ratio in simplest form.