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If ( A - B ) A + ² C ² ~A =1, ~A , ~B , C (0, 2 ) , then

Options

  1. AA , C , B are in A.P.
  2. BA, B, C are in G.P.
  3. CA, C, B are in G.P.
  4. DA , B , C are in A.P.

Correct answer

C. A, C, B are in G.P.

Step-by-step solution

Starting with (A-B) A + ^2 C ^2 A = 1 , rearrange to (A-B) A = 1 - ^2 C ^2 A . (A-B) A = ^2 A - ^2 C . Applying the identity ^2 = ^2 1+ ^2 and simplifying leads to a relationship between the tangent values. The constraint reduces to ( C)^2 = A B , which means A , C , and B form a geometric progression with common ratio r = C A .

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