Question

if the expression ,a²(b²-c²)x²+b²(c²-a²)x+c²(a²-b²) is a perfect square ,then show that a,b,c are in H.P progression

Answer 1

Given (a²-b²)³+(b²-c²)³+(c²-a²)³=3(a+b)(b+c)(c+a)(a-b)(b-c)(c-a)
Consider LHS:(a²-b²)³+(b²-c²)³+(c²-a²)³
If x + y + z =0 then x3 + y3 + z3 = 3xyz as,
Here x = a²-b², y = b²-c² and z = c²-a²
a²-b² + b²-c² + c²-a² = 0
Hence (a²-b²)³+(b²-c²)³+(c²-a²)³ = 3(a²-b²)(b²-c²)(c²-a²)³
                                                  = 3(a + b)(a − b)(b + c)(b − c)(c + a)(c − a)
                                                  = 3(a + b)(b + c)(c + a)(a − b)(b − c)(c − a)