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A plane which bisects the angle between ...

A plane which bisects the angle between the two given planes `2x-y+2z-4=0" and "x+2y+2z-2=0`, passes through the point

A

(1, -4, 1)

B

(1, 4, -1)

C

(2, 4, 1)

D

(2, -4, 1)

Text Solution

Verified by Experts

The correct Answer is:
D
Equation of planes bisecting the angles between the planes
`a_(1)x+b_(1)y+c_(1)z+d_(1)=0" and"`
`a_(2)x+b_(2)y+c_(2)z+d_(2)=0," and"`
`(a_(1)x+b_(1)y+c_(1)z+d_(1))/(sqrt(a_(1)^(2)+b_(1)^(2)+c_(1)^(2)))=+-(a_(2)x+b_(2)y+c_(2)z+d_(2))/(sqrt(a_(2)^(2)+b_(2)^(2)+c_(2)^(2)))`
Equation of given planes are
`2x-y+2z-4=0" "...(i)`
and `" "z+2y+2z-2=0" "...(ii)`
Now, equation of planes bisecting the angles between the planes (i) and (ii) are
`(2x-y+2z-4)/(sqrt(4-1-4))=+-(x+2y+2z-2)/(sqrt(1+4+4))`
`implies" "2x-y+2z-4=+-(x+2y+2z-2)`
On taking (+ve) sign, we get a plane
`x-3y=2" "...(iii)`
On taking(-ve)sing, we get a plane
`3x+y+4z=6" "...(iv)`
Now from the given options, the point (2, -4, 1) satisfy the plane of angle bisector `3x+y+4z=6`
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