If a + b + c = 10, a2 + b2 + c2 = 38, what is the value of (a - b)2 + (b - c)2 + (c - a)2?
Show answer & explanation
Correct answer: C
Explanation
The correct answer is Option 3 i.e. 14.
Here, we have a + b + c = 10, a2 + b2 + c2 = 38
Now, we know that
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
⇒ 102 = 38 + 2 (ab + bc + ca)
⇒ 62 = 2(ab + bc + ca)
⇒ ab + bc + ca = 31
Now, we know that
a3 + b3 + c3 - 3abc = (a + b + c){(a2 + b2 + c2) - (ab + bc + ca)}
⇒ 10(38 - 31) = 10 × 7 = 70
Now,
a3 + b3 + c3 - 3abc = 1/2(a + b + c) [(a - b)2 + (b - c)2 + (c - a)2]
⇒ 70 = 1/2 × 10 [(a - b)2 + (b - c)2 + (c - a)2]
⇒ (a - b)2 + (b - c)2 + (c - a)2 = 14