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Question

The equation of the bisector of the obtuse angle between the planes 3x+4y−5z+1=0,5x+12y−13z=0 is

A
11x+4y3z=0
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B
14x8y+13=0
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C
2x+8y8z1=0
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D
13x7z+18=0
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Solution

The correct option is C 2x+8y8z1=0
Plane1 :3x+4y5z+1=0
Plane2 :5x+12y12z=0
let us construct a ||gm ABCD with AB & AD in direction of normal to plane & plane2 respectively.
AB=3^i+4^j5^kAD=5^i+12^j13^k
AC will be the acute angle bisector whereas BD will be in direction of obtuse angle bisector to the normals.
AC=AB+AD (by ||gm law of addition )
BD=ABAD (by law of addition)
BD=2^i8^j+8^k is the direction of the normal to the plane through obtuse angle bisector plane1 & plane2.
Equation of plane through the line of intersection of plane1 & plane2
(3x+4y5z+1)+λ(5x+12y13z)=0(3+5λ)x+(4+12λ)y+(513λ)+1=0
The above plane should be parallel to the plane formed as it is normal.
3+5λ2=4+12λ8=513λ8λ=1
The required plane is ,
2x8y+8z+1=02x+8y8z1=0

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