GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 17 Jan 2020, 14:16 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # Every point in the xy plane satisfying the condition ax + by ≥ c is sa

Author Message
TAGS:

### Hide Tags

Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9983
Location: Pune, India
Every point in the xy plane satisfying the condition ax + by ≥ c is sa  [#permalink]

### Show Tags

2
9 00:00

Difficulty:   95% (hard)

Question Stats: 36% (02:05) correct 64% (02:21) wrong based on 190 sessions

### HideShow timer Statistics

Let's try this relatively simple question:

Every point in the xy plane satisfying the condition ax + by ≥ c is said to be in region R. If a, b and c are real numbers, does any point of region R lie in the third quadrant?

(1) Slope of the line represented by ax + by – c = 0 is 2.
(2) The line represented by ax + by – c = 0 passes through (-3, 0).

_________________
Karishma
Veritas Prep GMAT Instructor

Manager  Joined: 19 Aug 2010
Posts: 59
Re: Every point in the xy plane satisfying the condition ax + by ≥ c is sa  [#permalink]

### Show Tags

1
Quote:
2. The line represented by ax + by – c = 0 passes through (-3, 0).

with that information we don't know the slope of the line. It could be posivite, negative or a zero slope. If it has either positive or negative some solutions will lie in the III quadrant, if it has a zero slope-no. That is why I thought it's not sufficient.
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9983
Location: Pune, India
Re: Every point in the xy plane satisfying the condition ax + by ≥ c is sa  [#permalink]

### Show Tags

1
1
rockroars: I appreciate the effort for the diagrams. But you made a tiny judgment error. Let me explain the answer in detail.

First of all, notice that ax + by – c = 0 or ax + by = c is the equation of the same line. A line divides the plane into two regions. One of them, where every point (x, y) satisfies ax + by ≥ c, is region R.

Statement 1: Slope of line is 2
Attachment: Ques1.jpg [ 7.64 KiB | Viewed 5020 times ]

the line will pass through third quadrant and hence both regions will lie in the third quadrant. Sufficient.

Statement 2: The line passes through (-3, 0).
Attachment: Ques2.jpg [ 7.56 KiB | Viewed 5018 times ]

A line passing through (-3, 0) could be the blue line or the green line. In either case, the line will pass through the third quadrant and hence, will have both regions in the third quadrant. So it is sufficient too? What about the x axis? That is also a line passing through (-3, 0). It does not pass through the third quadrant. We would need the equation of the line to find out whether our region R lies in the third quadrant. The equation of x axis is y = 0. So the required region is y ≥ 0 i.e. the first and second quadrant. Hence using just this information, we cannot say whether a point of region R lies in the third quadrant or not.
_________________
Karishma
Veritas Prep GMAT Instructor

Manager  Status: I rest, I rust.
Joined: 04 Oct 2010
Posts: 93
Schools: ISB - Co 2013
WE 1: IT Professional since 2006
Re: Every point in the xy plane satisfying the condition ax + by ≥ c is sa  [#permalink]

### Show Tags

1
VeritasPrepKarishma wrote:
Let's try this relatively simple question:

Every point in the xy plane satisfying the condition ax + by ≥ c is said to be in region R. If a, b and c are real numbers, does any point of region R lie in the third quadrant?
1. Slope of the line represented by ax + by – c = 0 is 2.
2. The line represented by ax + by – c = 0 passes through (-3, 0).

Any line with a positive slope will pass from 1st and 3rd quadrant, and either "one of the other two quadrants" or the "origin".

S1: Slope is positive. Sufficient.
S2: Line passes from (-3,0) so the line either passes from 3rd Quadrant or the equation is of X-axis. Not Sufficient.

_________________
Respect,
Vaibhav

PS: Correct me if I am wrong.
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9983
Location: Pune, India
Re: Every point in the xy plane satisfying the condition ax + by ≥ c is sa  [#permalink]

### Show Tags

1
Rajkiranmareedu wrote:
Karishma: I didn't understand. Could you explain a bit.

VeritasPrepKarishma wrote:
rockroars: I appreciate the effort for the diagrams. But you made a tiny judgment error. Let me explain the answer in detail.

First of all, notice that ax + by – c = 0 or ax + by = c is the equation of the same line.A line divides the plane into two regions. One of them, where every point (x, y) satisfies ax + by ≥ c, is region R.

Statement 1: Slope of line is 2
Attachment:
Ques1.jpg

the line will pass through third quadrant and hence both regions will lie in the third quadrant. Sufficient.
Statement 2: The line passes through (-3, 0).
Attachment:
Ques2.jpg

A line passing through (-3, 0) could be the blue line or the green line. In either case, the line will pass through the third quadrant and hence, will have both regions in the third quadrant. So it is sufficient too? What about the x axis? That is also a line passing through (-3, 0). It does not pass through the third quadrant. We would need the equation of the line to find out whether our region R lies in the third quadrant. The equation of x axis is y = 0. So the required region is y ≥ 0 i.e. the first and second quadrant. Hence using just this information, we cannot say whether a point of region R lies in the third quadrant or not.

Check out this post. It discusses this question in detail: http://www.veritasprep.com/blog/2010/12 ... s-part-ii/

A line (which by definition, extends infinitely at both sides) given by ax+by - c = 0 splits a region into two sections - one on the left side of the line and the other on the right side of the side. One of these two regions will satisfy ax+by - c < 0 and the other will satisfy ax+by - c > 0.
How do you know which region satisfies which inequality? Put a point from that region in the inequalities and see what it satisfies e.g. if (0, 0) doesn't lie on the line, put x = 0, y = 0
If c is negative, ax+by - c < 0 will be satisfied and the region that contains the point (0, 0) i.e. the origin of the axis will satisfy ax+by - c < 0.
In that case, the other region will satisfy ax+by - c > 0.

Which quadrants will lie in any particular region depends on where the line is located. If it is a vertical line passing through first and fourth quadrant, second and third quadrant will lie to its left and some part of first and fourth quadrants will also lie to its left. Rest of the first and fourth quadrants will lie to its right (make a line and see what i mean)
Similarly, try making a line with a positive slope, say 2. It will pass through first and third quadrants in ALL cases (remember, a line extends indefinitely at both ends). Since it will pass through the third quadrant, both the regions will have a part of the third quadrant.
_________________
Karishma
Veritas Prep GMAT Instructor

Manager  Joined: 19 Aug 2010
Posts: 59
Re: Every point in the xy plane satisfying the condition ax + by ≥ c is sa  [#permalink]

### Show Tags

Is it A ?
I think A is sufficient becase if the slope is 2, then some solutions lie in the III quadrant.
B seems to me insufficient because I need at least the coordinates of 1 more point to find the slope.
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9983
Location: Pune, India
Re: Every point in the xy plane satisfying the condition ax + by ≥ c is sa  [#permalink]

### Show Tags

medanova wrote:
Is it A ?
I think A is sufficient becase if the slope is 2, then some solutions lie in the III quadrant.
B seems to me insufficient because I need at least the coordinates of 1 more point to find the slope.

Your answer is correct. Good! Though, I am not convinced with the reasoning. Think a little more...
_________________
Karishma
Veritas Prep GMAT Instructor

Retired Moderator Joined: 02 Sep 2010
Posts: 716
Location: London
Re: Every point in the xy plane satisfying the condition ax + by ≥ c is sa  [#permalink]

### Show Tags

A : Sufficient ... Since slope is 2, hence positive. This means that the line must pass through the third quadrant. The inequality ax+by>=c represents one side of the line. Since the line passes through the 3rd quadrant, either side has points from the 3rd quadrant. Hence the region will always havea bit of third quadrant.

B : Insuffcient ... Any line except the one with slope=0 will pass through 3rd quadrant and the above logic applies. But the line with slope 0 may or may not have the thrid quadrant points in the region included depending on which side we choose (sign of c)

_________________
Intern  Joined: 18 Mar 2010
Posts: 28
Re: Every point in the xy plane satisfying the condition ax + by ≥ c is sa  [#permalink]

### Show Tags

I think the answer should be C.

We get the details of line and regions completely only after we club 1 and 2. After which we will be in a position to decide if Region R passes through 3rd Quad.
Attachments Untitled.jpg [ 39.86 KiB | Viewed 5010 times ]

Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9983
Location: Pune, India
Re: Every point in the xy plane satisfying the condition ax + by ≥ c is sa  [#permalink]

### Show Tags

shrouded1 wrote:
A : Sufficient ... Since slope is 2, hence positive. This means that the line must pass through the third quadrant. The inequality ax+by>=c represents one side of the line. Since the line passes through the 3rd quadrant, either side has points from the 3rd quadrant. Hence the region will always havea bit of third quadrant.

B : Insuffcient ... Any line except the one with slope=0 will pass through 3rd quadrant and the above logic applies. But the line with slope 0 may or may not have the thrid quadrant points in the region included depending on which side we choose (sign of c)

I am used to perfect answers from you shrouded1... But I think you missed out on a point here. c = 0 we know because it has to be x axis since it passes through (-3, 0). Since y >= 0 is the first and second quadrant hence we know that the region R may or not lie in third quadrant. If instead, we had ax+by <=c, statement 2 would be sufficient too.

If I am missing something here, let me know. (I would like to believe that I didn't err in a question I made myself!)
_________________
Karishma
Veritas Prep GMAT Instructor

Intern  Joined: 18 Mar 2010
Posts: 28
Re: Every point in the xy plane satisfying the condition ax + by ≥ c is sa  [#permalink]

### Show Tags

@VeritasPrepKarishma:

Okay! I realized where I made a mistake, a line with slope 2 is acute to x-axis. I assumed it to be obtuse to X axis.

Thanks a lot for your reply Intern  Joined: 20 Apr 2013
Posts: 17
Concentration: Finance, Finance
GMAT Date: 06-03-2013
GPA: 3.3
WE: Accounting (Accounting)
Re: Every point in the xy plane satisfying the condition ax + by ≥ c is sa  [#permalink]

### Show Tags

Karishma: I didn't understand. Could you explain a bit.

VeritasPrepKarishma wrote:
rockroars: I appreciate the effort for the diagrams. But you made a tiny judgment error. Let me explain the answer in detail.

First of all, notice that ax + by – c = 0 or ax + by = c is the equation of the same line.A line divides the plane into two regions. One of them, where every point (x, y) satisfies ax + by ≥ c, is region R.

Statement 1: Slope of line is 2
Attachment:
Ques1.jpg

the line will pass through third quadrant and hence both regions will lie in the third quadrant. Sufficient.
Statement 2: The line passes through (-3, 0).
Attachment:
Ques2.jpg

A line passing through (-3, 0) could be the blue line or the green line. In either case, the line will pass through the third quadrant and hence, will have both regions in the third quadrant. So it is sufficient too? What about the x axis? That is also a line passing through (-3, 0). It does not pass through the third quadrant. We would need the equation of the line to find out whether our region R lies in the third quadrant. The equation of x axis is y = 0. So the required region is y ≥ 0 i.e. the first and second quadrant. Hence using just this information, we cannot say whether a point of region R lies in the third quadrant or not.
Manager  Joined: 21 Jan 2010
Posts: 234
Re: Every point in the xy plane satisfying the condition ax + by ≥ c is sa  [#permalink]

### Show Tags

Let's try this relatively simple question:

Every point in the xy plane satisfying the condition ax + by ≥ c is said to be in region R. If a, b and c are real numbers, does any point of region R lie in the third quadrant?
1. Slope of the line represented by ax + by – c = 0 is 2.
2. The line represented by ax + by – c = 0 passes through (-3, 0).

This is bit tricky question.
You have to know that a line with positive slope makes an acute angle with the x axis or it always passes through 1 and 3rd quadrant. This is like a theorm. Helpful in some cases. Since Slope is +. It gives the soln.

2.Now here you have look back at the question and see the value of a,b and c. The points a,b and c will always pass through 3rd quadrant barring one case. In case the line is x - axis itself. ie x = 0 . and c =0 and b = 1.

Hence A gives a clear soln.
Intern  Joined: 20 Apr 2013
Posts: 17
Concentration: Finance, Finance
GMAT Date: 06-03-2013
GPA: 3.3
WE: Accounting (Accounting)
Re: Every point in the xy plane satisfying the condition ax + by ≥ c is sa  [#permalink]

### Show Tags

Non-Human User Joined: 09 Sep 2013
Posts: 13968
Re: Every point in the xy plane satisfying the condition ax + by ≥ c is sa  [#permalink]

### Show Tags

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________ Re: Every point in the xy plane satisfying the condition ax + by ≥ c is sa   [#permalink] 28 Nov 2019, 08:06
Display posts from previous: Sort by

# Every point in the xy plane satisfying the condition ax + by ≥ c is sa  