Timmimmit
In xy plane, line k passes through (1,1) and m through (1, -1). Are lines k and m perpendicular?
(1) k and m intersect at (1, -1)
(2) k intersects x –axis at (1, 0)
This question can be efficiently solved by plotting (imagining) the graph of the lines k and m, rather than trying to find out slopes of the lines algebraically.
To know the slope of a line, we should be able to 'fix' the line on the X-Y plane. And to fix a line we need at least two points that the line passes through. With this in mind, lets see what's given to us.
Question stem:
k passes through \((1,1)\), and
m passes through \((1,-1)\)
At this point none of the lines are fixed. Both of them can rotate \(360^{\circ}\) about the points they are passing through.
S1) This gives another point for line K.
So now we have two points that line k passes through- \((1,1)\) and \((1,-1)\). Now we can fix our line k. (at this point we realize that like k is parallel to Y-axis).
But with this, we still do not have line m fixed. We cannot comment about their angle. Hence
insufficient.
S2) This given yet another point that the line k passes through. This is not required as our like k is already fixed. All we needed was another point on line m to fix it. Since we do not have any info on line m and it can still rotate about the point \((1,-1)\) and can take any angle w.r.t. line k, this info is insufficient.
S1+S2) As we do not have any new info regarding line m and we cant fix it yet, this is again
insufficient.
Hence, the answer is E.
Hope that helps.