In the x-y plane, PQ is a straight line. The coordinate of point P is

Question

In the x-y plane, PQ is a straight line. The coordinate of point P is (a, b), where a > 0. Does point Q lie in the third quadrant?

(1) The origin bisects PQ in 1: 1 ratio.
(2) The slope m of the line PQ is non-negative.

​​​​​​​    e-GMAT          Question of the Week #39

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CAtluri · Answer

IMO E

Both statements do not indicate anything. It could also be that the line can be just the X axis

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leomanna10 · Answer

IMO C

We know that P has a>0, so point P lie or in the first quadrant or in the fourth quadrant.

1) statment 1 not sufficient, because it tells us that the origin bisects the segment PQ in two equal segments. But Point Q can lie either in the second or in the third quadrant;

2) statment 2 not sufficent. We can only say that point Q cannot be in the second quadrant.

1+2) sufficient.

from 1: origin bisects PQ with ratio 1:1
from 2: the slope isn't negative, so Q can't be in the second quadrant.

IMO P lie in the first quadrant and Q in the third.

Hope my answer is correct.

paolodeppa · Answer

1) Only tells us that x-axis will cut the line in 2, but still don't know anything about P and Q positions
2) Only tells us that the line is either horizontal or increasing

1)+2) still ambiguous about which one between P or Q is at the bottom or top of x-axis

EgmatQuantExpert · Answer

Solution

Given:

• PQ is a straight line.
• Coordinate of point P is (a, b).
• a > 0.

To find:

• In which quadrant Q lies.

Let us assume the coordinate of point Q is (x, y)
To find the quadrant of point Q, we need to find the exact value of the coordinate of point Q.

Analysing statement 1:

The origin bisects PQ in 1: 1 ratio.
Hence, 0 =  \frac{a + x}{2} and 0=  \frac{b + y}{2} 
• Therefore. x= -a and y= -b
• However, we still don’t know the exact value of x and y.

Hence, statement 1 is not sufficient to answer the question.

Analysing statement 2:

The slope m of the line PQ is non-negative.
Slope m of a line is  \frac{y2 – y1}{x2-x1} .
Hence, m=  \frac{y-b}{x-a} 
However, we still don’t have any information about the values of x and y.
Hence, statement 2 is not sufficient to answer the question.

Combining both the statements together:

From statement 1:
• x= -a and y= -b
From statement 2:
• m=  \frac{y-b}{x-a} 
We cannot find the value of x and y even after combining both the statements together.
Hence, option E is the correct answer.

Correct answer: option E.

aryarkrishnan · Answer

EgmatQuantExpert   Bunuel

Can you let me know where I have gone wrong in my reasoning. 
 
I thought C is the answer and my reasoning is given below.

Given a straight line PQ, with co-ordinates of P as (a,b) where a > 0
 
=> P is either on the 1st quadrant or on x axis or on the 4rth quadrant. Any pattern of line with positive, negative, infinite or zero slope can be drawn using points in these quadrants.

STATEMENT 1 : Origin bisects the line in the ratio 1 : 1

From the set of lines that could possibly be the answer we can eliminate all lines that doesnt pass through the origin.

As line passes through origin and a >0, line y axis and all lines with slope infinity (|| to y axis) can be eliminated.

If point P is in quadrant 1 it has to pass though quadrant 3 to satisfy this condition. This implies that the line will have a positive slope

If b is 0, then PQ is part of the x axis itself, and the point Q will be (-a,0) => Slope is 0 here

If the point P lies in 4rth quadrant, then Q must lie in the 2nd quadrant so that the origin bisects PQ. Here PQ will have negative slope.

HENCE STATEMENT 1 is not enough to answer the question

STATEMENT 2 : slope is non negative.

0 is neither positive nor negative. So I believe this statement essentially means slope is positive. We cannot narrow down the options here. The point Q could be in first quadrant itself

However, combining the 2 we can conclude that Q should pass through origin and should be in third quadrant. Hence both together can answer the question

Saby1098 · Answer

Given a > 0. Means P is either in Quadrant 1 or Quadrant 4.

Statement 1: Origin bisects line PQ in the ratio 1:1 
Then if P is in 4th Quadrant, Q is in 2nd Quadrant 
and if P is in 1st Quadrant, Q is in 3rd Quadrant

INSUFF

Statement 2: Slope of line is non-negative, i.e m>=0
i.e the line definitely passes through 1st and 3rd Quadrant. However, it says nothing if Q is in the 1st or 3rd Quadrant

INSUFF

St 1 + St 2 
In this case, we know that the line passes through the origin and P&Q are on the opposite sides of the origin. 
Case 1: m>0, Given P is in the 1st Quadrant, Q is in the 3rd Quadrant 
Case 2: m =0. In this case P is on the right side of the Y axis and Q is on the left side of the Y axis and both P,Q are on the X axis. Its not in any quadrant

INSUFF

Hence E

Nidhijain94 · Answer

> Non-negative numbers mean numbers that are greater than or equal to zero. zero is called a non-negative number as well as non-positive. so you have to consider the slope m=0 as well.

ProKiller · Answer

Let a=2.

So P is (2,b). Now depending on b, P can either be in 1st quadrant, on x-axis, or in 4th quadrant. Since neither statement gives me any info about b, OR about b and point Q together, both statements are inconclusive. Option E.

DHRJ0032 · Answer

Such kind of questions can be solved very quickly(<100 secs) when we use graphical approach.you can now observe how easy it is this way !!!

E is the answer.

Posted from my mobile device

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In the x-y plane, PQ is a straight line. The coordinate of point P is

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In the x-y plane, PQ is a straight line. The coordinate of point P is

Prep Toolkit

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