If the square of the sum of two numbers is equal to 4 times of their product. then the ratio of these numbers is : (a) \(2: 1\) (b) \(1: 3\) (c) \(1: 1\) (d) \(1: 2\)

To find the ratio of two numbers given that the square of their sum is equal to four times their product, let’s denote the two numbers as \(a\) and \(b\). According to the given condition, we have:

\[
(a + b)^2 = 4ab
\]

Expanding the left side of the equation gives:

\[
a^2 + 2ab + b^2 = 4ab
\]

Rearranging the terms to bring them all to one side:

\[
a^2 + 2ab – 4ab + b^2 = 0
\]

Simplifying:

\[
a^2 – 2ab + b^2 = 0
\]

Notice that the left side of the equation now represents the square of the difference between \(a\) and \(b\):

\[
(a – b)^2 = 0
\]

For a square to equal zero, the quantity being squared must itself be zero:

\[
a – b = 0
\]

This implies:

\[
a = b
\]

Therefore, the ratio of \(a\) to \(b\) is \(1:1\), which means the correct answer is:

(c) \(1: 1\)