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Easy

Question

If a,b,c are in A.P then c o t A over 2 comma c o t B over 2 a n d space c o t C over 2  are in

  1. A.P
  2. G.P
  3. H.P
  4. A.G.P

Hint:

As we know that when a, b, c is in AP. Then 2b= a+ c. Also, to find the relation between c o t A over 2 comma c o t B over 2 a n d space c o t C over 2 first we will have to find the value of the three in terms of a, b and c then after looking at the values we can determine the relation.

The correct answer is: A.P


    Given a, b, c is in AP.
    So, b-a=c-b or, 2b=a+ c
    also, a=b-d, c=b +d, d is the common difference
    Now, we know that a+ b+ c=2srightwards double arrows=fraction numerator a plus b plus c over denominator 2 end fractionrightwards double arrow s equals fraction numerator 3 b over denominator 2 end fraction
    Now, cot A over 2 equals square root of fraction numerator s left parenthesis s minus a right parenthesis over denominator left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end fraction end root                            cotB over 2 equals square root of fraction numerator s left parenthesis s minus b right parenthesis over denominator left parenthesis s minus a right parenthesis left parenthesis s minus c right parenthesis end fraction end root                                                 cotC over 2 equals square root of fraction numerator s left parenthesis s minus c right parenthesis over denominator left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis end fraction end root
    equals square root of fraction numerator begin display style fraction numerator 3 b over denominator 2 end fraction end style open square brackets begin display style fraction numerator 3 b over denominator 2 end fraction end style minus open parentheses b minus d close parentheses close square brackets over denominator open square brackets begin display style fraction numerator 3 b over denominator 2 end fraction end style minus b close square brackets open square brackets begin display style fraction numerator 3 b over denominator 2 end fraction end style minus open parentheses b plus d close parentheses close square brackets end fraction end root
equals square root of fraction numerator begin display style fraction numerator 3 b over denominator 2 end fraction end style open parentheses begin display style fraction numerator 3 b minus 2 b plus 2 d over denominator 2 end fraction end style close parentheses over denominator open parentheses begin display style fraction numerator 3 b minus 2 b over denominator 2 end fraction end style close parentheses open parentheses begin display style fraction numerator 3 b minus 2 b minus 2 d over denominator 2 end fraction end style close parentheses end fraction end root
equals square root of fraction numerator begin display style fraction numerator 3 b over denominator 2 end fraction end style open parentheses begin display style fraction numerator b plus 2 d over denominator 2 end fraction end style close parentheses over denominator begin display style b over 2 end style open parentheses begin display style fraction numerator b minus 2 d over denominator 2 end fraction end style close parentheses end fraction end root
equals square root of fraction numerator 3 left parenthesis b plus 2 d right parenthesis over denominator open parentheses b minus 2 d close parentheses end fraction end root                              equals square root of fraction numerator begin display style fraction numerator 3 b over denominator 2 end fraction end style open square brackets begin display style fraction numerator 3 b over denominator 2 end fraction end style minus b close square brackets over denominator open square brackets begin display style fraction numerator 3 b over denominator 2 end fraction end style minus open parentheses b minus d close parentheses close square brackets open square brackets begin display style fraction numerator 3 b over denominator 2 end fraction end style minus open parentheses b plus d close parentheses close square brackets end fraction end root
equals square root of fraction numerator begin display style fraction numerator 3 b over denominator 2 end fraction end style open square brackets begin display style fraction numerator 3 b minus 2 b over denominator 2 end fraction end style close square brackets over denominator open square brackets begin display style fraction numerator 3 b minus 2 b plus d over denominator 2 end fraction end style close square brackets open square brackets begin display style fraction numerator 3 b minus 2 b minus 2 d over denominator 2 end fraction end style close square brackets end fraction end root
equals square root of fraction numerator begin display style fraction numerator 3 b over denominator 2 end fraction b over 2 end style over denominator open parentheses begin display style fraction numerator b plus d over denominator 2 end fraction end style close parentheses open parentheses begin display style fraction numerator b minus 2 d over denominator 2 end fraction end style close parentheses end fraction end root
equals square root of fraction numerator 3 b squared over denominator open parentheses b plus d close parentheses open parentheses b minus 2 d close parentheses end fraction end root equals square root of fraction numerator 3 b squared over denominator b squared minus 4 d squared end fraction end root                            equals square root of fraction numerator begin display style fraction numerator 3 b over denominator 2 end fraction end style open square brackets begin display style fraction numerator 3 b over denominator 2 end fraction end style minus open parentheses b plus d close parentheses close square brackets over denominator begin display style fraction numerator 3 b over denominator 2 end fraction end style minus b open square brackets begin display style fraction numerator 3 b over denominator 2 end fraction end style minus open parentheses b minus d close parentheses close square brackets end fraction end root
equals square root of fraction numerator begin display style fraction numerator 3 b over denominator 2 end fraction end style open parentheses begin display style fraction numerator 3 b minus 2 b minus 2 d over denominator 2 end fraction end style close parentheses over denominator begin display style fraction numerator 3 b minus 2 b over denominator 2 end fraction end style open parentheses begin display style fraction numerator 3 b minus 2 b plus 2 d over denominator 2 end fraction end style close parentheses end fraction end root
equals square root of fraction numerator begin display style fraction numerator 3 b over denominator 2 end fraction end style open parentheses begin display style fraction numerator b minus 2 d over denominator 2 end fraction end style close parentheses over denominator begin display style b over 2 end style open parentheses begin display style fraction numerator b plus 2 d over denominator 2 end fraction end style close parentheses end fraction end root
equals square root of fraction numerator 3 open parentheses b minus 2 d close parentheses over denominator open parentheses b plus 2 d close parentheses end fraction end root
    n o w comma space c o t A over 2 plus c o t B over 2 equals square root of fraction numerator 3 open parentheses b plus 2 d close parentheses over denominator b minus 2 d end fraction end root plus square root of fraction numerator 3 open parentheses b minus 2 d close parentheses over denominator b plus 2 d end fraction end root
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals square root of 3 open square brackets fraction numerator square root of open parentheses b plus 2 d close parentheses squared end root plus square root of open parentheses b minus 2 d close parentheses squared end root over denominator square root of open parentheses b minus 2 d close parentheses open parentheses b plus 2 d close parentheses end root end fraction close square brackets
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals square root of 3 open parentheses fraction numerator b plus 2 d plus b minus 2 d over denominator square root of b squared minus 4 d squared end root end fraction close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals square root of 3 open parentheses fraction numerator 2 b over denominator square root of b squared minus 4 d squared end root end fraction close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 2 open parentheses fraction numerator square root of 3 b squared end root over denominator square root of b squared minus 4 d squared end root end fraction close parentheses space equals 2 c o t B over 2
t h e r e f o r e comma space c o t A over 2 comma space c o t B over 2 space a n d space c o t C over 2 space i s space i n space A. P.

space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space

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