Is |a-c| + |a| = |c|? (1) ab > bc (2) ab < 0

We can solve most absolute value equation type questions with the square approach taught by Aditya Kumar video @ GMATClub Youtube channel.

|a-c| = |c| - |a|

Since We have at least one absolute value statement in either left or right of the equation. We can easily square them on both sides.

(a-c) ^2 = | (c|- |a|) ^2
a^2 + c^2 - 2ac = |c|^2 + |a|^2 - 2|a||c|
(Since a^2>= 0 we can cancel out a^2 and c^2 with corresponding square of their absolute values)
-2ac = - 2|a||c|
ac = |a||c|

Since Right hand side is positive left part ac is only positive when a and care of same sign. OR ac>= 0
Also, we have |a - c| = |c| - |a|..Since both parts are positive, we must have |c| > |a|

Statement 1: ab > bc..a>c or a <c depends on the sign of b NOT SUFFICIENT
Statement 2: ab<0..a and b are of opposite signs..we can not infer anything..NOT SUFFICIENT

Combined:
ab<0 And ab > bc i.e. 0>ab>bc

If b = +ve, then 0>a>c (ALSO a and c have to be -ve or ab or bc to be less than 0)
a>c when both -ve satisfies..Both of same sign also |c| > |a|..As 'a' being greater in the -ve part of the number line entails |c| > |a|

If b = -ve, then c>a and both a and c positive
It also satisifies same sign also |c| > |a|

So 'C' is the answer