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AP EAMCET2005MathematicsProperties of Triangles

aligned & In A B C , & a ( ^2 B+ ^2 C )+ A(c C+b B) is & equal to aligned

Options

  1. Aa
  2. Bb
  3. Cc
  4. Da+b+c

Correct answer

A. a

Step-by-step solution

aligned & a ( ^2 B+ ^2 C )+ A(c C+b B) & =a [ ( a^2+c^2-b^2 2 a c )+ ( a^2+b^2-c^2 2 a b ) ] & + b^2+c^2-a^2 2 b c [c ( b^2+a^2-c^2 2 a b )+b ( a^2+c^2-b^2 2 a c ) ] & = (a^2+c^2-b^2 )^2 4 a c^2 + (a^2+b^2-c^2 )^2 4 a b^2 & + (b^2+c^2-a^2 ) (b^2+a^2-c^2 ) 4 a b^2 & + (b^2+c^2-a^2 ) (a^2+c^2-b^2 ) 4 a c^2 aligned aligned & = (a^2+c^2-b^2 ) (a^2+c^2-b^2+b^2+c^2-a^2 ) 4 a c^2 & + (a^2+b^2-c^2 ) (a^2+b^2-c^2+b^2+c^2-a^2 ) 4 a b^2 & = a^2+c^2-b^2 2 a + a^2+b^2-c^2 2 a = 2 a^2 2 a =a aligned

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