Let equation of the required line be
== ......(i)
Given two lines
== .......(ii)
and
== ......(iii)
Since the line (i) is perpendicular to both the lines (ii) and (iii), therefore
a−b+c=0 ......(iv)
−2b+c=0 .....(v)
From (iv) and (v)
c=2b and
a+b=0, which are not satisfy by options (c) and (d). Hence options (c) and (d) are rejected.
Thus point
(x1,y1,z1) on the required line will be either (0,0,0) or (1,-1,0).
Now foot of the perpendicular from point (0,0,0) to the line(iii)
=(1,−2r−1,r)
The direction ratios of the line joining the points (0,0,0) and
(1,−2r−1,r) are
1,−2r−1,r Since sum of the
x and
y -coordinate of direction ratio of the required line is 0 .
∴1−2r−1=0,⇒r=0 Hence direction ratio are 1,-1,0
But the z-direction ratio of the required line is twice the
y -direction ratio of the required line
i.e.
0=2(−1), which is not true.
Hence the shortest line does not pass through the point (0,0,0) . Therefore option (a) is also rejected.