To determine the ratio of the areas of two circles given a specific scale factor, we need to understand how scaling affects the area of a circle. Specifically, if the linear dimensions (like radius) of a circle are scaled by a factor, the area of the circle scales by the square of that factor.
Let's break it down step-by-step:
1. Given scale factor: The scale factor between the two circles is [tex]\(\frac{2x}{5y}\)[/tex].
2. Understanding the effect on areas: The area of a circle is proportional to the square of its radius. This means if the radius of the second circle is [tex]\(\frac{2x}{5y}\)[/tex] times the radius of the first circle, then the area of the second circle is [tex]\(\left(\frac{2x}{5y}\right)^2\)[/tex] times the area of the first circle.
3. Calculate the ratio of the areas:
[tex]\[
\text{Ratio of the areas} = \left(\frac{2x}{5y}\right)^2 = \frac{(2x)^2}{(5y)^2} = \frac{4x^2}{25y^2}.
\][/tex]
Thus, the ratio of their areas is [tex]\(\frac{4x^2}{25y^2}\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{4 x^2}{25 y^2}}
\][/tex]
