The C.G. of the shaded area of the below figure whose curve OM is a parabola from y-axis, is A. $$\frac{{\text{a}}}{4}$$ B. $$\frac{{3{\text{a}}}}{4}$$ C. $$\frac{{3{\text{b}}}}{{10}}$$ D. $$\frac{{3{\text{a}}}}{{10}}$$ E. $$\frac{{3{\text{a}}}}{5}$$

The correct answer is $\boxed{\frac{{3{\text{a}}}}{5}}$.

The centroid of a parabola is located at its $y$-coordinate, which is also the average of the $y$-coordinates of the endpoints of the parabola’s base. In this case, the parabola’s base is the line segment $AB$, whose endpoints have $y$-coordinates $a$ and $-a$. Therefore, the centroid of the shaded area is located at $\frac{{a} – {(-a)}}{2} = \boxed{\frac{{3{\text{a}}}}{5}}$.

Here is a more detailed explanation of each option:

• Option A: $\frac{{\text{a}}}{4}$. This is the $y$-coordinate of the point $C$, which is the midpoint of the line segment $AB$. However, the centroid is not located at the midpoint of the base of the parabola, so this is not the correct answer.
• Option B: $\frac{{3{\text{a}}}}{4}$. This is the $y$-coordinate of the point $D$, which is the midpoint of the line segment $CD$. However, the centroid is not located at the midpoint of any of the line segments that make up the parabola, so this is not the correct answer.
• Option C: $\frac{{3{\text{b}}}}{{10}}$. This is the $y$-coordinate of the point $E$, which is the midpoint of the line segment $EF$. However, the centroid is not located at the midpoint of any of the line segments that make up the parabola, so this is not the correct answer.
• Option D: $\frac{{3{\text{a}}}}{{10}}$. This is the $y$-coordinate of the point $F$, which is the midpoint of the line segment $FG$. However, the centroid is not located at the midpoint of any of the line segments that make up the parabola, so this is not the correct answer.
• Option E: $\frac{{3{\text{a}}}}{5}$. This is the $y$-coordinate of the point $G$, which is the centroid of the parabola. Therefore, this is the correct answer.