Question

If x⁴ + y⁴ + x²y² = 117, and x² + y² - xy = 9, then find the value of (x² + y²) × xy.
1. 15
2. 30
3. 34
4. 22

Answer 1

Correct Answer - Option 4 : 22

Given:

x⁴ + y⁴ + x²y² = 117

x² + y² - xy = 9

Formula Used:

(x² + y² - xy)(x² + y² + xy) = x⁴ + y⁴ + x²y²

Calculation:

(x² + y² - xy)(x² + y² + xy) = x⁴ + y⁴ + x²y²

⇒ 9(x² + y² + xy) = 117

⇒ (x² + y² + xy) = 13      ---(1)

(x² + y² - xy) = 9      ---(2)

Subtract eq. (2) from eq. (1)

⇒ (x² + y² + xy) - (x² + y² - xy) = 13 - 9

⇒ 2xy = 4

⇒ xy = 2

Now,

Add eq. (1) and eq. (2)

⇒ (x² + y² + xy) + (x² + y² - xy) = 13 + 9

⇒ 2(x² + y²) = 22

⇒ x² + y² = 11

⇒ x² + y² = 11

(x² + y²) × xy = 11 × 2

⇒ 22

∴ The value of (x² + y²) × xy is 22.