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Which of the following equation represent pair of perpendicular planes?
- a)2x2 - 2y2 + 4z2 - 2yz+ 6xz - 3xy = 0
- b)6x2+4y2-10z2+3yz+4zx-11xy=0
- c)12x2-y2+6z2+7yz+6zx+2xy=0
- d)12x2-y2+2z2-yz+5zx+xy=0
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
Which of the following equation represent pair of perpendicular planes...
Planes represented by (b) are perpendicular planes.
Proof: The planes are given by
6x2+4y2-10z2+3yz+4zx-11xy=0
∴ a=6, b=4, c=-10
∴ a+b+c=6-4-10=0
Remark. Two planes arc perpendicular if the sum of the coefficients of x2, y2 and z2 is zero.
Proof: The planes are given by
6x2+4y2-10z2+3yz+4zx-11xy=0
∴ a=6, b=4, c=-10
∴ a+b+c=6-4-10=0
Remark. Two planes arc perpendicular if the sum of the coefficients of x2, y2 and z2 is zero.
Most Upvoted Answer
Which of the following equation represent pair of perpendicular planes...
Planes represented by (b) are perpendicular planes.
Proof: The planes are given by
6x2+4y2-10z2+3yz+4zx-11xy=0
∴ a=6, b=4, c=-10
∴ a+b+c=6-4-10=0
Remark. Two planes arc perpendicular if the sum of the coefficients of x2, y2 and z2 is zero.
Proof: The planes are given by
6x2+4y2-10z2+3yz+4zx-11xy=0
∴ a=6, b=4, c=-10
∴ a+b+c=6-4-10=0
Remark. Two planes arc perpendicular if the sum of the coefficients of x2, y2 and z2 is zero.
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Community Answer
Which of the following equation represent pair of perpendicular planes...
Perpendicular Planes:
Perpendicular planes are two planes that intersect each other at a right angle (90 degrees). In order to determine if a pair of planes are perpendicular, we need to examine the coefficients of the variables in the equations of the planes.
Equation Analysis:
Let's analyze the given equations to determine which pair of planes are perpendicular:
a) 2x^2 - 2y^2 + 4z^2 - 2yz + 6xz - 3xy = 0
b) 6x^2 + 4y^2 - 10z^2 + 3yz + 4zx - 11xy = 0
c) 12x^2 - y^2 + 6z^2 + 7yz + 6zx + 2xy = 0
d) 12x^2 - y^2 + 2z^2 - yz + 5zx + xy = 0
Analysis:
- For two planes to be perpendicular, the dot product of their normal vectors should be zero.
- The normal vectors are given by the coefficients of x, y, and z terms in the equations.
- Let's find the normal vectors for each pair of planes:
a) Normal vector: (2, -3, 6)
b) Normal vector: (6, -11, 4)
c) Normal vector: (12, 2, 6)
d) Normal vector: (12, 1, 5)
Conclusion:
- The pair of planes represented by equation b) 6x^2 + 4y^2 - 10z^2 + 3yz + 4zx - 11xy = 0 have normal vectors (6, -11, 4) and are perpendicular since the dot product of their normal vectors is zero.
- Therefore, option b) represents a pair of perpendicular planes.
Perpendicular planes are two planes that intersect each other at a right angle (90 degrees). In order to determine if a pair of planes are perpendicular, we need to examine the coefficients of the variables in the equations of the planes.
Equation Analysis:
Let's analyze the given equations to determine which pair of planes are perpendicular:
a) 2x^2 - 2y^2 + 4z^2 - 2yz + 6xz - 3xy = 0
b) 6x^2 + 4y^2 - 10z^2 + 3yz + 4zx - 11xy = 0
c) 12x^2 - y^2 + 6z^2 + 7yz + 6zx + 2xy = 0
d) 12x^2 - y^2 + 2z^2 - yz + 5zx + xy = 0
Analysis:
- For two planes to be perpendicular, the dot product of their normal vectors should be zero.
- The normal vectors are given by the coefficients of x, y, and z terms in the equations.
- Let's find the normal vectors for each pair of planes:
a) Normal vector: (2, -3, 6)
b) Normal vector: (6, -11, 4)
c) Normal vector: (12, 2, 6)
d) Normal vector: (12, 1, 5)
Conclusion:
- The pair of planes represented by equation b) 6x^2 + 4y^2 - 10z^2 + 3yz + 4zx - 11xy = 0 have normal vectors (6, -11, 4) and are perpendicular since the dot product of their normal vectors is zero.
- Therefore, option b) represents a pair of perpendicular planes.
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