Question

If $x^2 + y^2 = 25$ and $xy = 12$, then what is the value of $x^4 + y^4$? 

• A. 337
• B. 193
• C. 289
• D. 241

Hint:

• Use the algebraic identity: \((a+b)^2 = a^2 + b^2 + 2ab\).
• Let \(a = x^2\) and \(b = y^2\). Then, \((x^2 + y^2)^2 = x^4 + y^4 + 2x^2 y^2\).
• Rearrange to find \(x^4 + y^4\): \[ x^4 + y^4 = (x^2 + y^2)^2 - 2(xy)^2. \]
• Substitute the given values \(x^2 + y^2 = 25\) and \(xy = 12\) to compute the result.

Answer 1

The Correct Option is A

We are given the equations: \[ x^2 + y^2 = 25 \quad \text{and} \quad xy = 12. \] We need to find the value of \( x^4 + y^4 \). Recall the algebraic identity: \[ (x^2 + y^2)^2 = x^4 + y^4 + 2x^2 y^2. \] Rearranging this, we get: \[ x^4 + y^4 = (x^2 + y^2)^2 - 2(xy)^2. \] Substitute the given values: \[ x^2 + y^2 = 25, \quad xy = 12 \implies (xy)^2 = 12^2 = 144. \] Therefore, \[ x^4 + y^4 = 25^2 - 2 \times 144 = 625 - 288 = 337. \] Hence, the required value is: \[ \boxed{337}. \]