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Question

Line a x plus b y plus p equals 0 makes angle pi divided by 4 with x blank c o s blank alpha plus y blank s i n blank alpha equals p comma p space element of R to the power of straight prime If these lines and the line x blank s i n blank alpha minus y blank c o s blank alpha equals 0 are concurrent, then

  1. a to the power of 2 end exponent plus b to the power of 2 end exponent equals 1    
  2. a to the power of 2 end exponent plus b to the power of 2 end exponent equals 2    
  3. 2 left parenthesis a to the power of 2 end exponent plus b to the power of 2 end exponent right parenthesis equals 1    
  4. a to the power of 2 end exponent minus b to the power of 2 end exponent equals 1    

The correct answer is: a to the power of 2 end exponent plus b to the power of 2 end exponent equals 2


    We should find some relation between a and b using given conditions
    Lines xcosα+ysinα=p andxsinαycosα=0 are mutually perpendicular. Thus ax+by+p=0will be equally inclined to these lines and would be the angle bisector of these lines. Now equations of angle bisectors is
    xsinαycosα=±(xcosα+ysinαp)
     x(cosαsinα)+y(cosα+sinα)=p
    or
     x(sinα+cosα)y(cosαsinα)=p
    Comparing these lines withax+by+p=0, we get
     cosαsinαa=sinα+cosαb=1
    a2+b2=2
    or
    cosα+sinαa=sinα+cosαb=1
    a2+b2=2

    Hence option 2 is correct

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