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In the xy-plane, line l passes through the points (0,2) and (1,0), and 09 May 2018, 02:04
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In the xy-plane, line l passes through the points (0,2) and (1,0), and line m passes through the points (0,-4) and (4,0). If (a,b) is the point of intersection of line l and line m, which of the following is true?

a) a > 0 and b > 0
b) a > 0 and b < 0
c) a < 0 and b > 0
d) a < 0 and b < 0
e) a = 0 or b = 0
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B
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In the xy-plane, line l passes through the points (0,2) and (1,0), and 10 May 2018, 10:41
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alanforde800Maximus
In the xy-plane, line l passes through the points (0,2) and (1,0), and line m passes through the points (0,-4) and (4,0). If (a,b) is the point of intersection of line l and line m, which of the following is true?

a) a > 0 and b > 0
b) a > 0 and b < 0
c) a < 0 and b > 0
d) a < 0 and b < 0
e) a = 0 or b = 0
Show more

We can begin by finding an equation for each line:

Line l:

slope = (0-2)/(1-0) = -2, y-intercept = (0, 2) = 2

Thus, an equation for line l is y = -2x + 2

Line m:

Slope = (0-(-4)/(4-0) = 1, y-intercept = (0, -4) = -4

Thus, an equation for line m is y = x - 4.

Now, let’s find their intersection by setting the right hand side of the two equations equal to each other:

-2x + 2 = x - 4

-3x = -6

x = 2

Substitute x = 2 back to either equation (let’s take the one for line m), we have:

y = 2 - 4 = -2

We see that the point of intersection is (2, -2), so a = 2 > 0 and b = -2 < 0.

Alternate Solution:

Using the given points, make a quick sketch of the two lines on the same set of coordinate axes. Note that line l has a negative slope and line m has a positive slope. Their point of intersection will be in Quadrant 4, where x is always positive and y is always negative.

Answer: B
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In the xy-plane, line l passes through the points (0,2) and (1,0), and 09 May 2018, 02:38
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alanforde800Maximus
In the xy-plane, line l passes through the points (0,2) and (1,0), and line m passes through the points (0,-4) and (4,0). If (a,b) is the point of intersection of line l and line m, which of the following is true?

a) a > 0 and b > 0
b) a > 0 and b < 0
c) a < 0 and b > 0
d) a < 0 and b < 0
e) a = 0 or b = 0
Show more


The answer is B.

If you draw a line in XY plane, you can see they intersect in Quadrant-IV.

So, a>0 and b<0

Attachment:
File comment: XY Plane
Capture.JPG
Capture.JPG [ 13.24 KiB | Viewed 19927 times ]
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In the xy-plane, line l passes through the points (0,2) and (1,0), and 09 May 2018, 18:05
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alanforde800Maximus
In the xy-plane, line l passes through the points (0,2) and (1,0), and line m passes through the points (0,-4) and (4,0). If (a,b) is the point of intersection of line l and line m, which of the following is true?

a) a > 0 and b > 0
b) a > 0 and b < 0
c) a < 0 and b > 0
d) a < 0 and b < 0
e) a = 0 or b = 0


The answer is B.

If you draw a line in XY plane, you can see they intersect in Quadrant-IV.

So, a>0 and b<0

Attachment:
Capture.JPG
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Thanks for the graphical solution. However, is there any algebraic approach as well that can be deployed to solve this question?
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In the xy-plane, line l passes through the points (0,2) and (1,0), and 12 Apr 2020, 23:58
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Graphical approach is recommended for this question as it is easy and short

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alanforde800Maximus
In the xy-plane, line l passes through the points (0,2) and (1,0), and line m passes through the points (0,-4) and (4,0). If (a,b) is the point of intersection of line l and line m, which of the following is true?

a) a > 0 and b > 0
b) a > 0 and b < 0
c) a < 0 and b > 0
d) a < 0 and b < 0
e) a = 0 or b = 0


The answer is B.

If you draw a line in XY plane, you can see they intersect in Quadrant-IV.

So, a>0 and b<0

Attachment:
Capture.JPG

Thanks for the graphical solution. However, is there any algebraic approach as well that can be deployed to solve this question?
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In the xy-plane, line l passes through the points (0,2) and (1,0), and 03 Aug 2020, 05:43
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alanforde800Maximus
In the xy-plane, line l passes through the points (0,2) and (1,0), and line m passes through the points (0,-4) and (4,0). If (a,b) is the point of intersection of line l and line m, which of the following is true?

a) a > 0 and b > 0
b) a > 0 and b < 0
c) a < 0 and b > 0
d) a < 0 and b < 0
e) a = 0 or b = 0

We can begin by finding an equation for each line:

Line l:

slope = (0-2)/(1-0) = -2, y-intercept = (0, 2) = 2

Thus, an equation for line l is y = -2x + 2

Line m:

Slope = (0-(-4)/(4-0) = 1, y-intercept = (0, -4) = -4

Thus, an equation for line m is y = x - 4.

Now, let’s find their intersection by setting the right hand side of the two equations equal to each other:

-2x + 2 = x - 4

-3x = -6

x = 2

Substitute x = 2 back to either equation (let’s take the one for line m), we have:

y = 2 - 4 = -2

We see that the point of intersection is (2, -2), so a = 2 > 0 and b = -2 < 0.

Alternate Solution:

Using the given points, make a quick sketch of the two lines on the same set of coordinate axes. Note that line l has a negative slope and line m has a positive slope. Their point of intersection will be in Quadrant 4, where x is always positive and y is always negative.

Answer: B
Show more

Hi Scott

Could you please elaborate on the alternate solution a bit more? A line that has a negative slope and a line that has a positive slope will always intersect in Quad 4? Not sure ive understood your alternate solution properly. Thanks in advance
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In the xy-plane, line l passes through the points (0,2) and (1,0), and 03 Aug 2020, 12:25
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alanforde800Maximus
In the xy-plane, line l passes through the points (0,2) and (1,0), and line m passes through the points (0,-4) and (4,0). If (a,b) is the point of intersection of line l and line m, which of the following is true?

a) a > 0 and b > 0
b) a > 0 and b < 0
c) a < 0 and b > 0
d) a < 0 and b < 0
e) a = 0 or b = 0

We can begin by finding an equation for each line:

Line l:

slope = (0-2)/(1-0) = -2, y-intercept = (0, 2) = 2

Thus, an equation for line l is y = -2x + 2

Line m:

Slope = (0-(-4)/(4-0) = 1, y-intercept = (0, -4) = -4

Thus, an equation for line m is y = x - 4.

Now, let’s find their intersection by setting the right hand side of the two equations equal to each other:

-2x + 2 = x - 4

-3x = -6

x = 2

Substitute x = 2 back to either equation (let’s take the one for line m), we have:

y = 2 - 4 = -2

We see that the point of intersection is (2, -2), so a = 2 > 0 and b = -2 < 0.

Alternate Solution:

Using the given points, make a quick sketch of the two lines on the same set of coordinate axes. Note that line l has a negative slope and line m has a positive slope. Their point of intersection will be in Quadrant 4, where x is always positive and y is always negative.

Answer: B

Hi Scott

Could you please elaborate on the alternate solution a bit more? A line that has a negative slope and a line that has a positive slope will always intersect in Quad 4? Not sure ive understood your alternate solution properly. Thanks in advance
Show more

Great question. By chance did you actually make a sketch of the given lines? If not, can you, and then let me know if the solution makes sense?
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In the xy-plane, line l passes through the points (0,2) and (1,0), and 03 Aug 2020, 22:22
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alanforde800Maximus
In the xy-plane, line l passes through the points (0,2) and (1,0), and line m passes through the points (0,-4) and (4,0). If (a,b) is the point of intersection of line l and line m, which of the following is true?

a) a > 0 and b > 0
b) a > 0 and b < 0
c) a < 0 and b > 0
d) a < 0 and b < 0
e) a = 0 or b = 0

We can begin by finding an equation for each line:

Line l:

slope = (0-2)/(1-0) = -2, y-intercept = (0, 2) = 2

Thus, an equation for line l is y = -2x + 2

Line m:

Slope = (0-(-4)/(4-0) = 1, y-intercept = (0, -4) = -4

Thus, an equation for line m is y = x - 4.

Now, let’s find their intersection by setting the right hand side of the two equations equal to each other:

-2x + 2 = x - 4

-3x = -6

x = 2

Substitute x = 2 back to either equation (let’s take the one for line m), we have:

y = 2 - 4 = -2

We see that the point of intersection is (2, -2), so a = 2 > 0 and b = -2 < 0.

Alternate Solution:

Using the given points, make a quick sketch of the two lines on the same set of coordinate axes. Note that line l has a negative slope and line m has a positive slope. Their point of intersection will be in Quadrant 4, where x is always positive and y is always negative.

Answer: B

Hi Scott

Could you please elaborate on the alternate solution a bit more? A line that has a negative slope and a line that has a positive slope will always intersect in Quad 4? Not sure ive understood your alternate solution properly. Thanks in advance

Great question. By chance did you actually make a sketch of the given lines? If not, can you, and then let me know if the solution makes sense?
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Hey,

I tried making a negative sloping and a positive sloping line in every quadrant. I guess it can intersect in any quadrant you want it to.. but, basically if your x intercept is +ve then your y intercept will also be +ve and similarly for the opposite sign..
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In the xy-plane, line l passes through the points (0,2) and (1,0), and 19 Sep 2021, 11:40
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Given: In the xy-plane, line l passes through the points (0,2) and (1,0), and line m passes through the points (0,-4) and (4,0).

Asked: If (a,b) is the point of intersection of line l and line m, which of the following is true?

Equation of line l: -
y - 0 = (2-0)/(0-1) (x-1)
y = -2(x-1)

Equation of line m:-
y-0 = (-4-0)/(0-4) (x-4)
y = (x-4)

Point of Intersection of line l and line m:-
y = - 2(x-1) = x-4
-2x + 2 = x - 4
3x = 6
x = a = 2 > 0
y = b = -2 < 0


IMO B
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In the xy-plane, line l passes through the points (0,2) and (1,0), and 09 Aug 2025, 12:43
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