Question

Prove that : (a + b {)}^{2} \: + (b + c {)}^{2} \: - (c + a {)}^{2}(a+b) 2 +(b+c) 2 −(c+a) 2 = 2( {b}^{2} + ab + bc - ca)2(b 2 +ab+bc−ca) ​

Answer 1

Step-by-step explanation:

Solution:-

On taking LHS

(a+b)²+(b+c)²-(c+a)²

=> (a²+2ab+b²)+(b²+2bc+c²)-(c²+2ca+a²)

=> a²+2ab+b²+b²+2bc+c²-c²-2ca-a²

=> (a²-a²)+(b²+b²)+(c²-c²)+(2ab+2bc-2ca)

=> 0+2b²+0+2ab+2bc-2ca

=> 2b²+2ab+2bc-2ca

=> 2(b²+ab+bc-ca)

=>RHS

=> LHS = RHS

Therefore,

(a+b)²+(b+c)²-(c+a)² = 2(b²+ab+bc-ca)

Hence, Proved.

Used formulae:-

→ (x+y)² = x²+2xy+y²