Q The number of distinct real values of λ for which the lines x-11 = y-22=z+3λ2 and x-31=y-2λ2 = - Maths - Three Dimensional Geometry

Question

Q The number of distinct real values of λ for which the lines
x - 1 1   =   y - 2 2 = z + 3 λ 2   a n d
  x - 3 1 = y - 2 λ 2   =   z - 1 2   a r
e   c o p l a n e r   i s
(1) 4 (2) 1 (3) 2 (4) 3

Neha Sethi · Accepted Answer

Dear student
We know that if the line x-x1l1=y-y1m1=z-z1n1 andx-x2l2=y-y2m2=z-z2n2 are coplanar then,x2-x1y2-y1z2-z1l1m1n1l2m2n2=0Here x1=1,y1=2,z1=-3l1=1,m1=2,n1=λ2x2=3,y2=2,z2=1l2=1,m2=λ2,n2=2So, we have 3-12-21+312λ21λ22=020412λ21λ22=0⇒24-λ4-0+4λ2-2=0⇒8-2λ4+4λ2-8=0⇒λ4-2λ2=0⇒λ2λ2-2=0either λ2=0 or λ2-2=0⇒λ=0,λ2=2So, λ=0,±2So, correct option  is 4
Regards

Q. The number of distinct real values of λ for which the lines x - 1 1 = y - 2 2 = z + 3 λ 2 a n d x - 3 1 = y - 2 λ 2 = z - 1 2 a r e c o p l a n e r i s (1) 4 (2) 1 (3) 2 (4) 3

Q. The number of distinct real values of $λ$ for which the lines $\frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z + 3}{λ^{2}} a n d \frac{x - 3}{1} = \frac{y - 2}{λ^{2}} = \frac{z - 1}{2} a r e c o p l a n e r i s$
(1) 4 (2) 1 (3) 2 (4) 3