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A(4,4) and B(-4,4) are two points on the circle x^2+y^2=32 and P is any point on the circle.Prove that the angular bisector of angle APB pass through a fixed point.

A(4,4) and B(-4,4) are two points on the circle x^2+y^2=32 and P is any point on the circle.Prove that the angular bisector of angle APB pass through a fixed point.

Grade:Upto college level

1 Answers

Badiuddin askIITians.ismu Expert
148 Points
12 years ago

Dear sudarshan


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angular bisector of angel must made 45/2 angel with each line AP and BP

 we know

  tan 45 = 2tan45/2 /(1- tan245/2)

let tan 45/2  = m

so 1= 2m/(1-m2)

 m2 +2m -1 =0 ................1

 

now slope of line AP (m1) = (4- 4√2sinθ)/((4- 4√2cosθ)

 slope of bisector m2 = (y- 4√2sinθ)/((x- 4√2cosθ)

 

 so tan45/2 =m= (m1-m2)/(1+m1m2)

 put value of m1 and m2 and simplify

√2 cosθ ( -mx-4m +y -4) + √2sinθ (-my -4m+4-x) +(mx +my +32m -y +x) =0

 solve coefficiet of cosθ and sinθ  and find value of x and y and put it in constant term if it satiesfied it then this line always pass throug this point for all value of θ.

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Regards,
Askiitians Experts
Badiuddin



 

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