find the value of a b + BC + CA if a + b + C is equal to 9 and a square + b square + c square is equal to 35
Answers
Answer:
a² + b² + c² = ab + bc + ca
On multiplying both sides by “2”, it becomes
2 ( a² + b² + c² ) = 2 ( ab + bc + ca)
2a² + 2b² + 2c² = 2ab + 2bc + 2ca
a² + a² + b² + b² + c² + c² – 2ab – 2bc – 2ca = 0
a² + b² – 2ab + b² + c² – 2bc + c² + a² – 2ca = 0
(a² + b² – 2ab) + (b² + c² – 2bc) + (c² + a² – 2ca) = 0
(a – b)² + (b – c)² + (c – a)² = 0
=> Since the sum of square is zero then each term should be zero
⇒ (a –b)² = 0, (b – c)² = 0, (c – a)² = 0
⇒ (a –b) = 0, (b – c) = 0, (c – a) = 0
⇒ a = b, b = c, c = a
∴ a = b = c.
Step-by-step explanation:
GIVEN:-
=> a + b + c = 9
=> a² + b² + c² = 35
To find :-
ab + bc + ca = ?
Solution:-
Use this identity
=> ( a + b + c )² = a² + b² + c² + 2 ( ab + bc + ca )
Use the given value
a + b + c = 9
a² + b² + c² = 35 , we get
=> ( 9 )² = 35 + 2 ( ab + bc + ca )
=> 81 = 35 + 2 ( ab + bc + ca )
=> 81 - 35 = 2 ( ab + bc + ca )
=> 46 = 2 ( ab + bc + ca )
=> ab + bc + ca = 46/ 2
=> ab + bc + ca = 23