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If line L has a positive slope, what is the x-intercept of L

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by GMATGuruNY » Mon Sep 02, 2013 9:11 pm
If line L in the xy-coordinate plane has a positive slope, what is the x-intercept of L ?

(1) There are different points (a, b) and (c, d) on line L such that ad = bc.

(2) There are constants m and n such that the points (m, n) and (-m, -n) are both on line L
Many slope questions can be solved efficiently by DRAWING.

Statement 1: There are different points (a, b) and (c, d) on line L such that ad = bc.
Case 1: a=-1, d=4, b=-2, c=2
Here, line L contains (a,b) = (-1,-2) and (c,d) = (2,4):
Image
Since line L passes through the origin, its x-intercept is 0.

Case 2: a=-4, d=2, b=-1, c=8
Here, line L contains (a,b) = (-4,-1) and (c,d) = (8,2):
Image
Since line L passes through the origin, its x-intercept is 0.

The two random cases above illustrate that -- when ad=bc -- line L must pass through the origin and thus will have an x-intercept of 0.
SUFFICIENT.

Statement 2: There are constants m and n such that the points (m, n) and (-m, -n) are both on line L
Case 3: m=1 and n=1
Here, line L contains (m,n) = (1,1) and (-m,-n) = (-1,-1):
Image
Since line L passes through the origin, its x-intercept is 0.

Case 4: m=-3, n=-3
Here, line L contains (m,n) = (-3,-3) and (-m,-n) = (3,3):
Image
Since line L passes through the origin, its x-intercept is 0.

The two random cases above illustrate that -- since Line L contains both (m,n) and (-m,-n) -- it must pass through the origin and thus will have an x-intercept of 0.
SUFFICIENT.

The correct answer is D.
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by kevincanspain » Tue Sep 03, 2013 3:51 am
Mitch's method is fantastic!

Also, for (1), you could reason as follows:


Any two different points on the line can be written as (a,b) and (c,d)=(a+k, b+mk), where m is the slope of the line, and k is a positive number.

That ad=bc tells us that ab +amk = ba + bk

Thus amk =bk , meaning that am=b.
So (a,b) = (a,am)
If (x,0) is also on this line with slope m, (am)/(a-x) = m ,and assuming that a is not equal to x, a= a - x, and therefore x=0. Thus the x-intercept has to be 0, which is to say that the line passes through the origin

If a=x, (a,b) and (x,0) are the same point, and the line passes through the origin.

In any event, the x-intercept is 0.

Perhaps there is a shorter way to arrive at this conclusion mathematically, but NO way will be more efficient that Mitch's! It's amazing how many questions are best done by means of examples
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by kevincanspain » Tue Sep 03, 2013 3:51 am
Mitch's method is fantastic!

Also, for (1), you could reason as follows:


Any two different points on the line can be written as (a,b) and (c,d)=(a+k, b+mk), where m is the slope of the line, and k is a positive number.

That ad=bc tells us that ab +amk = ba + bk

Thus amk =bk , meaning that am=b.
So (a,b) = (a,am)
If (x,0) is also on this line with slope m, (am)/(a-x) = m ,and assuming that a is not equal to x, a= a - x, and therefore x=0. Thus the x-intercept has to be 0, which is to say that the line passes through the origin

If a=x, (a,b) and (x,0) are the same point, and the line passes through the origin.

In any event, the x-intercept is 0.

Perhaps there is a shorter way to arrive at this conclusion mathematically, but NO way will be more efficient that Mitch's! It's amazing how many questions are best done by means of examples
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by Mission2012 » Tue Sep 03, 2013 4:24 am
Hi Mitch,

I had a little different approach for (1) -

as (a,b) and (c,d) are different points on the line and ad = bc or b/a = d/c

As for any point (m,n) the slope of line joining the point and origin will have slope = n/m
Hence slope of these two points with line joining them to origin b/a and d/c respectively.

As b/a = d/c, it implies both the point are collinear with origin. Hence x intercept it 0

Is this approach right?
GMATGuruNY wrote:
If line L in the xy-coordinate plane has a positive slope, what is the x-intercept of L ?

(1) There are different points (a, b) and (c, d) on line L such that ad = bc.

(2) There are constants m and n such that the points (m, n) and (-m, -n) are both on line L
Many slope questions can be solved efficiently by DRAWING.

Statement 1: There are different points (a, b) and (c, d) on line L such that ad = bc.
Case 1: a=-1, d=4, b=-2, c=2
Here, line L contains (a,b) = (-1,-2) and (c,d) = (2,4):
Image
Since line L passes through the origin, its x-intercept is 0.

Case 2: a=-4, d=2, b=-1, c=8
Here, line L contains (a,b) = (-4,-1) and (c,d) = (8,2):
Image
Since line L passes through the origin, its x-intercept is 0.

The two random cases above illustrate that -- when ad=bc -- line L must pass through the origin and thus will have an x-intercept of 0.
SUFFICIENT.

Statement 2: There are constants m and n such that the points (m, n) and (-m, -n) are both on line L
Case 3: m=1 and n=1
Here, line L contains (m,n) = (1,1) and (-m,-n) = (-1,-1):
Image
Since line L passes through the origin, its x-intercept is 0.

Case 4: m=-3, n=-3
Here, line L contains (m,n) = (-3,-3) and (-m,-n) = (3,3):
Image
Since line L passes through the origin, its x-intercept is 0.

The two random cases above illustrate that -- since Line L contains both (m,n) and (-m,-n) -- it must pass through the origin and thus will have an x-intercept of 0.
SUFFICIENT.

The correct answer is D.
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by faraz_jeddah » Tue Sep 03, 2013 9:53 pm
GMATGuruNY wrote:

Statement 1: There are different points (a, b) and (c, d) on line L such that ad = bc.
Case 1: a=-1, d=4, b=-2, c=2
Here, line L contains (a,b) = (-1,-2) and (c,d) = (2,4):
Image
Since line L passes through the origin, its x-intercept is 0.

Case 2: a=-4, d=2, b=-1, c=8
Here, line L contains (a,b) = (-4,-1) and (c,d) = (8,2):
Image
Since line L passes through the origin, its x-intercept is 0.

The two random cases above illustrate that -- when ad=bc -- line L must pass through the origin and thus will have an x-intercept of 0.
SUFFICIENT.

.
Mitch,

For evaluating Statement 1 - How would you know that the line passes through the origin? You would have to find that out using equation of a line which you havent mentioned in your solution.
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by GMATGuruNY » Tue Sep 03, 2013 10:07 pm
faraz_jeddah wrote:
For evaluating Statement 1 - How would you know that the line passes through the origin? You would have to find that out using equation of a line which you havent mentioned in your solution.
Connect (-1,-2) and (2.4), and you get a line that passes through the origin.
Connect (-4,-1) and (8,2), and you get a line that passes through the origin.
There is no need to determine the equation of either line.
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by faraz_jeddah » Tue Sep 03, 2013 10:16 pm
GMATGuruNY wrote:
faraz_jeddah wrote:
For evaluating Statement 1 - How would you know that the line passes through the origin? You would have to find that out using equation of a line which you havent mentioned in your solution.
Connect (-1,-2) and (2.4), and you get a line that passes through the origin.
Connect (-4,-1) and (8,2), and you get a line that passes through the origin.
There is no need to determine the equation of either line.
Dont you think that would leave chance for an error? I mean I just drew a rough graph and I cannot be 100% sure that the x int would be 0. It appears as 0.5 as my grids are not accurate.
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by GMATGuruNY » Tue Sep 03, 2013 11:41 pm
faraz_jeddah wrote:
GMATGuruNY wrote:
faraz_jeddah wrote:
For evaluating Statement 1 - How would you know that the line passes through the origin? You would have to find that out using equation of a line which you havent mentioned in your solution.
Connect (-1,-2) and (2.4), and you get a line that passes through the origin.
Connect (-4,-1) and (8,2), and you get a line that passes through the origin.
There is no need to determine the equation of either line.
Dont you think that would leave chance for an error? I mean I just drew a rough graph and I cannot be 100% sure that the x int would be 0. It appears as 0.5 as my grids are not accurate.
Even a rudimentary drawing should be sufficient to see that (a.b) and (c,d) exhibit SYMMETRY about the origin.

Case 1:
(-1,-2) --> left 1, down 2.
(2, 4) --> right 2, up 4.
For each point, the horizontal distance from the origin is 1/2 the vertical distance from the origin.

Case 2:
(-4, -1) --> left 4, down 1.
(8,2) --> right 8, up 2.
For each point, the horizontal distance from the origin is 4 times the vertical distance from the origin.

Since in each case (a,b) and (c,d) exhibit symmetry about the origin, the line that connects them must pass THROUGH the origin.
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by gmattesttaker2 » Thu Sep 05, 2013 9:03 pm
GMATGuruNY wrote:
If line L in the xy-coordinate plane has a positive slope, what is the x-intercept of L ?

(1) There are different points (a, b) and (c, d) on line L such that ad = bc.

(2) There are constants m and n such that the points (m, n) and (-m, -n) are both on line L
Many slope questions can be solved efficiently by DRAWING.

Statement 1: There are different points (a, b) and (c, d) on line L such that ad = bc.
Case 1: a=-1, d=4, b=-2, c=2
Here, line L contains (a,b) = (-1,-2) and (c,d) = (2,4):
Image
Since line L passes through the origin, its x-intercept is 0.

Case 2: a=-4, d=2, b=-1, c=8
Here, line L contains (a,b) = (-4,-1) and (c,d) = (8,2):
Image
Since line L passes through the origin, its x-intercept is 0.

The two random cases above illustrate that -- when ad=bc -- line L must pass through the origin and thus will have an x-intercept of 0.
SUFFICIENT.

Statement 2: There are constants m and n such that the points (m, n) and (-m, -n) are both on line L
Case 3: m=1 and n=1
Here, line L contains (m,n) = (1,1) and (-m,-n) = (-1,-1):
Image
Since line L passes through the origin, its x-intercept is 0.

Case 4: m=-3, n=-3
Here, line L contains (m,n) = (-3,-3) and (-m,-n) = (3,3):
Image
Since line L passes through the origin, its x-intercept is 0.

The two random cases above illustrate that -- since Line L contains both (m,n) and (-m,-n) -- it must pass through the origin and thus will have an x-intercept of 0.
SUFFICIENT.

The correct answer is D.
Hello Mitch,

Thank you very much for the thorough and excellent explanation.

Best Regards,
Sri
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by Java_85 » Sun Sep 08, 2013 10:09 am
IMO D it is.
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