Option 3 : 2

**Given:**

\( \frac{{{{\left( {987~+~513} \right)}^2}~+~{{\left( {987~-~513} \right)}^2}}}{{{{\left( {987} \right)}^2}~+~\;{{\left( {513} \right)}^2}}}\)

**Concept Used:**

(a + b)^{2} = (a^{2} + b^{2} + 2ab) ----(1)

(a - b)^{2} = (a2 + b2 - 2ab) ----(2)

**Calculation:**

Adding equation (1) and (2), we get

(a + b)2 + (a - b)2 = 2(a2 + b2)

Where,

a = 987 and b = 513

According to the question,

\(\frac{{{{\left( {987~+~513} \right)}^2}~+~{{\left( {987~-~513} \right)}^2}}}{{{{\left( {987} \right)}^2}~+~\;{{\left( {513} \right)}^2}}}~=~\;\frac{{2[{{\left({987}\right)}^2}\;~+~\;{{\left({513}\right)}^2}]}}{{{{\left[({987}\right)}^2}\;~+~\;{{\left({513}\right)}^2}]}}\)

⇒ 2

**∴ The value of \( \frac{{{{\left( {987~+~513} \right)}^2}~+~{{\left( {987~-~513} \right)}^2}}}{{{{\left( {987} \right)}^2}~+~\;{{\left( {513} \right)}^2}}}\) is 2.**