Maths-
General
Easy

Question

The line l is tangent to the circle S at the point A. B and C are points on l on opposite sides of A and other tangents from B. C to S intersect at a point P. If B, C vary a longblank l blank in such a way that the product vertical line A B vertical line times vertical line A C vertical line is constant, then locus of P is

  1. circle    
  2. a line parallel to BC    
  3. a set of points    
  4. None of these    

The correct answer is: a line parallel to BC



    angle P B A equals B comma angle P C A equals gamma
    A B minus P subscript 1 end subscript comma A C minus q subscript 1 end subscript
    p subscript 1 end subscript q subscript 1 end subscript equals k to the power of 2 end exponent
    table row cell text end text text L end text text e end text text t end text text end text S text end text text t end text text o end text text u end text text c end text text h end text text end text B P text end text text a end text text n end text text d end text text end text C P text end text text a end text text t end text text end text E text end text text. end text text end text end cell row cell l n invisible function application triangle P E I end cell end table
    angle E I P equals fraction numerator 1 over denominator 2 end fraction left parenthesis beta plus gamma right parenthesis
    t subscript 1 end subscript equals r t a n invisible function application fraction numerator left parenthesis beta plus gamma right parenthesis over denominator 2 end fraction equals fraction numerator open parentheses p subscript 1 end subscript plus q subscript 1 end subscript close parentheses r to the power of 2 end exponent over denominator p subscript 1 end subscript q subscript 1 end subscript minus r to the power of 2 end exponent end fraction
    text Semi-perimeter of  end text triangle B C P text  is end text
    p subscript 1 end subscript plus q subscript 1 end subscript plus t subscript 1 end subscript equals p subscript 1 end subscript plus q subscript 1 end subscript plus fraction numerator open parentheses p subscript 1 end subscript plus q subscript 1 end subscript close parentheses r to the power of 2 end exponent over denominator p subscript 1 end subscript q subscript 1 end subscript minus r to the power of 2 end exponent end fraction equals fraction numerator p subscript 1 end subscript q subscript 1 end subscript open parentheses p subscript 1 end subscript plus q subscript 1 end subscript close parentheses over denominator p subscript 1 end subscript q subscript 1 end subscript minus r to the power of 2 end exponent end fraction
    A r e a blank o f blank capital delta B C P equals fraction numerator 1 over denominator 2 end fraction open parentheses p subscript 1 end subscript plus q subscript 1 end subscript close parentheses P H equals fraction numerator r p subscript 1 end subscript q subscript 1 end subscript open parentheses p subscript 1 end subscript plus q subscript 1 end subscript close parentheses over denominator p subscript 1 end subscript q subscript 1 end subscript minus r to the power of 2 end exponent end fraction
    P H equals fraction numerator 2 k to the power of 2 end exponent r over denominator k to the power of 2 end exponent minus r end fraction
    So, the locus of P is a line parallel to BC

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