if a,b,c are in an AP and if the equations (b-c)x^2 + (c-a)x + (a-b)=0 and 2(c+a)x^2 + (b+c)x=0 HAVE A COMMON ROOT, then, prove that :-

a^2, c^2, b^2 are in AP.

[*x^2 refers to x square ]

hi shridhar,

well in eq.1, by observation we can say that x=1 is d root so a+b the odr root..

similarly in eq.2 x=0 is a root.. since common roots.. x=1 is also d soln of eq.2..

satisfting x=1 in eq.2 we get:

2a +3c +b =0

since a,b,c r in AP substituing a=b-d n c=b+d whre d is common difrence, we get

6b +d =0

hence a=7b =>a^2 =49b^2 similarly b^2 =25b^2

hence a^2 c^2 and b^2 r in AP wid common diff =24b^2