Question : If a( a + b + c) = 45, b( a + b + c) = 75, and c( a + b + c) =105, then what is the value of a2 + b2 + c2?
Option 1: 75
Option 2: 83
Option 3: 217
Option 4: 225
Correct Answer: 83
Solution :
a(a + b + c) = 45 ------------------------- (1)
b(a + b + c) = 75 ------------------------- (2)
c(a + b + c) = 105 ------------------------ (3)
Adding (1), (2), and (3), we get (a + b + c)
2
= 225
⇒ (a + b + c) = 15 or (a + b + c) = –15
Case 1: (a + b + c) = 15 -------------------(4)
Dividing (4) by (1), we get a = 3
Dividing (4) by (2), we get b = 5
Dividing (4) by (1), we get c = 7
(a
2
+ b
2
+ c
2
) = 3
2
+ 5
2
+ 7
2
= 83
Case 2: (a + b+ c) = –15 -------------(5)
Dividing (5) by (1), we get a = –3
Dividing (5) by (2), we get b = –5
Dividing (5) by (1), we get c = –7
(a
2
+ b
2
+ c
2
) = (–3)
2
+ (–5)
2
+ (–7)
2
= 83
Hence, the correct answer is 83.
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