Question:
If a + b + c = 4, (a + b)2 + (b + c)2 + (c + a)2 = 20. Find the value of (a2 + b2 + c2).
If a + b + c = 4, (a + b)2 + (b + c)2 + (c + a)2 = 20. Find the value of (a2 + b2 + c2).
Updated On: Aug 21, 2024
- 4
- 8
- 6
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The Correct Option is A
Solution and Explanation
a+b+c=4 {given}
Squaring both sides, we get
(a + b + c)2 = 42
Or, a2 + b2 + c2 + 2ab + 2bc + 2ca = 16 … (I)
(a + b)2 + (b + c)2 + (c + a)2 = 20 {given}
Or, {a2 + b2 + 2ab} + {b2 + c2 + 2bc} + {a2 + c2 + 2ca} = 20
Or, 2(a2 + b2 + c2) + 2ab + 2bc + 2ca = 20
Or, 2(a2 + b2 + c2) + 16 - a2 - b2 - c2 = 20 [using equation (I)]
Or, (a2 + b2 + c2) + 16 = 20
Or, (a2 + b2 + c2) = 4
So, the correct option is (A) : 4.
Squaring both sides, we get
(a + b + c)2 = 42
Or, a2 + b2 + c2 + 2ab + 2bc + 2ca = 16 … (I)
(a + b)2 + (b + c)2 + (c + a)2 = 20 {given}
Or, {a2 + b2 + 2ab} + {b2 + c2 + 2bc} + {a2 + c2 + 2ca} = 20
Or, 2(a2 + b2 + c2) + 2ab + 2bc + 2ca = 20
Or, 2(a2 + b2 + c2) + 16 - a2 - b2 - c2 = 20 [using equation (I)]
Or, (a2 + b2 + c2) + 16 = 20
Or, (a2 + b2 + c2) = 4
So, the correct option is (A) : 4.
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