If a + b = 13 and $$a^3+b^3=559$$​. Find the value of $$a^2+b^2 .$$​​

Correct option is B

Given:
a + b = 13
​$$a^3 + b^3 = 559$$​​
Formula Used:
​$$a^3 + b^3 = (a+b)(a^2 + b^2 - ab)$$​​
​$$(a+b)^2 = a^2 + b^2 + 2ab$$​​
Solution:
Substitute the known values into the sum of cubes formula:
​$$559 = 13(a^2 + b^2 - ab)$$​​
​$$a^2 + b^2 - ab = \frac{559}{13} = 43$$​​
Substitute the known values into the square formula:
​$$13^2 = a^2 + b^2 + 2ab$$​​
​$$169 = a^2 + b^2 + 2ab$$​​
We now have two equations:
1)$$a^2 + b^2 - ab$$​ = 43
2)$$a^2 + b^2 + 2ab$$​ = 169
Subtract equation (1) from equation (2):
3ab = 169 - 43 = 126
ab = 42
Substitute the value of ab back into equation (1):
​$$a^2 + b^2 - 42 = 43$$​​
$$a^2 + b^2$$​= 43 + 42 = 85
Final Answer
So the correct answer is (b)