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Question

Vertices of a triangle are (4, 3) , left parenthesis 5 blank s i n blank capital theta comma blank 5 blank c o s blank capital theta right parenthesis comma left parenthesis 5 blank c o s blank capital theta comma blank minus 5 blank s i n blank capital theta right parenthesis Then the locus of the orthocenter is

  1. a straight line    
  2. a circle    
  3. ellipse    
  4. hyperbola    

The correct answer is: a circle


    Let A left parenthesis 43 right parenthesis comma B left parenthesis 5 blank s i n blank theta comma blank 5 blank c o s blank theta right parenthesis comma C left parenthesis 5 blank c o s blank theta comma blank minus 5 blank s i n blank theta right parenthesis be the vertices and G H be the centroid and orthocentre of capital delta A B C If O is the origin O A to the power of 2 end exponent equals 25 equals O B to the power of 2 end exponent equals O C to the power of 2 end exponent.
    O is the circumcentre of capital delta A B C times G divides OH in the ratio 1 : 2.
    Hence fraction numerator 4 plus 5 blank s i n blank theta plus 5 blank c o s blank theta over denominator 3 end fraction equals fraction numerator x subscript 1 end subscript over denominator 3 end fraction
    fraction numerator 3 plus 5 blank c o s blank theta minus 5 blank s i n blank theta over denominator 3 end fraction equals fraction numerator y subscript 1 end subscript over denominator 3 end fraction
    H is left parenthesis x subscript 1 end subscript comma y subscript 1 end subscript right parenthesis equals succeeds left parenthesis x subscript 1 end subscript minus 4 right parenthesis to the power of 2 end exponent equals left parenthesis y subscript 1 end subscript minus 3 right parenthesis to the power of 2 end exponent equals 50
    equals locus of H is left parenthesis x minus 4 right parenthesis to the power of 2 end exponent plus left parenthesis y minus 4 right parenthesis to the power of 2 end exponent equals 50 which is a circle.

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