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E
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Difficulty:
55%
(hard)
Question Stats:
56% (01:27) correct
44%
(01:34)
wrong
based on 232
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Line m passes through the origin. Line l is parallel to line m. What are the equations of the two lines?
(1) The horizontal distance between the two lines is 5 units.
(2) Line l has a y-intercept of 2.5
(1) The horizontal distance between the two lines is 5 units.
(2) Line l has a y-intercept of 2.5
19 Mar 2012, 01:06
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Line m passes through the origin. Line l is parallel to line m. What are the equations of the two lines?
Equation of a line in point intercept form is \(y=mx+b\), where: \(m\) is the slope of the line and \(b\) is the y-intercept of the line (the value of \(y\) for \(x=0\)).
From the stem:
Since parallel lines have the same slope, then the slopes of l and m are the same;
Since a line passing through the origin has y-intercept equal to zero then the equation of line m would be \(y_m=mx\) and the equation of line l would be \(y_l=mx+b\)
(1) The horizontal distance between the two lines is 5 units --> basically we are told that the x-intercept of line l is either -5 or 5, so we know that line l passes either through the point (-5, 0) or (5, 0). Not sufficient.
(2) Line l has a y-intercept of 2.5 --> \(b=2.5\), so we know that line l passes through the point (0, 2.5). One point is not enough to determine (fix) a line. Not sufficient.
(1)+(2) Now, even take together we cannot determine whether line l passes through the point (-5, 0) or (5, 0). So, we would have two possible points of line l: (-5, 0) and (0, 2.5) OR (5, 0) and (0, 2.5), which means that we would have two possible equations of line l and m: \(y_l=0.5*x+2.5\) and \(y_m=0.5*x\) OR \(y_l=-0.5*x+2.5\) and \(y_m=-0.5*x\). Not sufficient.
Answer: E.
To demonstrate.
Case 1: 
Case 1.png [ 7.73 KiB | Viewed 13769 times ]
Case 2: 
Case 2.png [ 8.08 KiB | Viewed 13755 times ]
For more on this subject please check Coordinate Geometry chapter of Math Book: math-coordinate-geometry-87652.html
Hope it helps.
Equation of a line in point intercept form is \(y=mx+b\), where: \(m\) is the slope of the line and \(b\) is the y-intercept of the line (the value of \(y\) for \(x=0\)).
From the stem:
Since parallel lines have the same slope, then the slopes of l and m are the same;
Since a line passing through the origin has y-intercept equal to zero then the equation of line m would be \(y_m=mx\) and the equation of line l would be \(y_l=mx+b\)
(1) The horizontal distance between the two lines is 5 units --> basically we are told that the x-intercept of line l is either -5 or 5, so we know that line l passes either through the point (-5, 0) or (5, 0). Not sufficient.
(2) Line l has a y-intercept of 2.5 --> \(b=2.5\), so we know that line l passes through the point (0, 2.5). One point is not enough to determine (fix) a line. Not sufficient.
(1)+(2) Now, even take together we cannot determine whether line l passes through the point (-5, 0) or (5, 0). So, we would have two possible points of line l: (-5, 0) and (0, 2.5) OR (5, 0) and (0, 2.5), which means that we would have two possible equations of line l and m: \(y_l=0.5*x+2.5\) and \(y_m=0.5*x\) OR \(y_l=-0.5*x+2.5\) and \(y_m=-0.5*x\). Not sufficient.
Answer: E.
To demonstrate.
Case 1:
Attachment:
Case 1.png [ 7.73 KiB | Viewed 13769 times ]
Attachment:
Case 2.png [ 8.08 KiB | Viewed 13755 times ]
Hope it helps.
General Discussion
18 Mar 2012, 23:50
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Smita04
Since the lines are parallel the slopes would be same, say m
Let the Line M be y=mx
So line L will be y=mx+c
Statement 1
Considering the horizontal distance on x axis , this distance can be taken towards +ve or -ve side of origin.
So the slopes of the lines and the value of c cannot be determined.
So insufficient.
Statement 2
Now we know the value of c but we still don't know the value of slope.
So insufficient.
From 1+2
We know that horizontal distance is 5 i.e x intercept and y intercept is 2.5 i.e c=2.5
Since the y intercept is +ve ,horizontal distance i.e x intercept is -ve.
So the line passes through (-5,0)
therefore using the equation we get
0=m(-5)+2.5
or m=1/2
So the equations are y=0.5x
and y=0.5x+2.5
So the answer is C
