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The planes bx - ay = n, cy - bz = l, az - cx = m intersect in a line if
- a)a+b+c=0
- b)a=b=c
- c)al+bm=cn=0
- d)l=m=n=0
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
The planes bx - ay = n, cy - bz = l, az - cx = m intersect in a line i...
The equation of the planes are
∴ Required conditions if that
al+bm+cn=0
∴ Required conditions if that
al+bm+cn=0
Most Upvoted Answer
The planes bx - ay = n, cy - bz = l, az - cx = m intersect in a line i...
Explanation:
To determine whether the planes intersect in a line, we need to check if the three given equations are consistent with each other.
Let's solve the given system of equations using the method of elimination:
1) bx - ay = n
2) cy - bz = l
3) az - cx = m
Step 1: Multiply equation 1 by c, equation 2 by b, and equation 3 by a:
1) bcx - acy = nc
2) bcy - bza = lb
3) aaz - acx = ma
Step 2: Add equations 1, 2, and 3 together:
bcx - acy + bcy - bza + aaz - acx = nc + lb + ma
Simplifying the equation:
(b - a)cx + (bc - a) y + (az - b) = nc + lb + ma
Step 3: Rearrange the equation:
(cx + y(b - a)) + (z(a - b)) = nc + lb + ma
Let's define two new variables:
p = cx + y(b - a)
q = z(a - b)
Now the equation becomes:
p + q = nc + lb + ma
Conclusion:
From the equation p + q = nc + lb + ma, we can see that the equation is satisfied when the coefficients of p and q are both zero. This means that the planes intersect in a line when:
a - b = 0 (equivalent to a = b)
and
a - b = 0 (equivalent to c = 0)
Therefore, the correct answer is option 'C': a = b = c = 0.
To determine whether the planes intersect in a line, we need to check if the three given equations are consistent with each other.
Let's solve the given system of equations using the method of elimination:
1) bx - ay = n
2) cy - bz = l
3) az - cx = m
Step 1: Multiply equation 1 by c, equation 2 by b, and equation 3 by a:
1) bcx - acy = nc
2) bcy - bza = lb
3) aaz - acx = ma
Step 2: Add equations 1, 2, and 3 together:
bcx - acy + bcy - bza + aaz - acx = nc + lb + ma
Simplifying the equation:
(b - a)cx + (bc - a) y + (az - b) = nc + lb + ma
Step 3: Rearrange the equation:
(cx + y(b - a)) + (z(a - b)) = nc + lb + ma
Let's define two new variables:
p = cx + y(b - a)
q = z(a - b)
Now the equation becomes:
p + q = nc + lb + ma
Conclusion:
From the equation p + q = nc + lb + ma, we can see that the equation is satisfied when the coefficients of p and q are both zero. This means that the planes intersect in a line when:
a - b = 0 (equivalent to a = b)
and
a - b = 0 (equivalent to c = 0)
Therefore, the correct answer is option 'C': a = b = c = 0.
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