Line L passes through points R(0, -5 ) and S(4,0) (see figure above).

24. Coordinate Geometry

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The equation of the line is\(\frac{x - x_1}{x_1 - x_2} = \frac{y - y_1}{y_1 - y_2}\)

\(\frac{x - 0}{0 - 4} = \frac{y + 5}{-5 - 0}\)

5x - 4y = 20

1) If x>4, 5x > 20. So, y must be greater than 0. Sufficient.

2) If y>-5, y can be 0 in which case x would be 4 or y can be 5 in which case x would be 8. So xy can be 0 or 40. Insufficient.

Answer is A.

Line L passes through points (0, -5) and S (4, 0) (see figure above). Point P with coordinates (x, y) is a point on line L. Is xy > 0 ?

(1) x > 4. Look at the diagram: since point P is on line L, and x > 4, then point P must be in the I quadrant, where both x and y coordinates of a point are positive. Thus xy > 0. Sufficient.

(2) y > -5. Again, from the diagram it follows that point P is either in the IV quadrant (here x is positive and y is negative --> xy < 0) or in the I quadrant (here both x and y are positive --> xy > 0). Not sufficient.

Answer: A.

Hope it's clear.
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Thanks a ton for your time Bunuel..

But it is still not clear to me, since x>4 , x could be in either first quadrant or fourth ,in first quadrant x and y are both positive
so xy>0 but in fourth quadrant x is positive and y negative so xy<0..

Since A is only telling about x and nothing about y how can we presume that y>0 as well so why A??

Look at the diagram below:Now, since X-coordinate of point P is more than 4 AND point P is on line L, then it must be on the red part of the line, where Y-coordinate is positive too.

Hope it's clear.
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For xy>0, x and y should have the same sign (positive X positive >0 and negative X negative >0) and not equal to 0.

In this case, that means that the point P should be in I or III quadrant.

1) x > 4

If x >4, then it should be right of the point S (4,0), and as the slope is positive, y will be > 0 and point P is in the first quadrant. Sufficient.

(2) y > - 5

If y > -5, point P is to the right of point R (0,-5), hence it can be in quadrants I and IV, if it's in I then xy >0, if it's in IV then xy <0 (because y<0 and x>0).

Answer

Question : is a point on Line L Is xy > 0?

for xy to be greater than zero, the sign of x and y both be same (either both positive or both Negative)

Sign of both quadrant is positive in 1st Quadrant
and Sign of both quadrant is Negative in 3rd Quadrant

Question REDEFINED : Does point P lie in the 1st Quadrant or 3rd Quadrant?

Statement 1: x > 4

For x to be greater than 4, the point P must lie on the line towards the right of point S. [As per the figure]
Towards the right of Point S, every part of the Line is in 1st Quadrant

Hence SUFFICIENT

Statement 2: y > - 5

For y to be Greater than -5, the point P must lie on the line towards the Right of point R. [As per the figure]
Towards the right of Point R, Some part of the Line is in 4th Quadrant and Some part is in 1st Quadrant

Hence NOT SUFFICIENT

Answer: option

From the diagram we can say that Line L passed from quadrant 1,3 and 4. If we know where (x,y) lies we would be able to say x*y>0 or not.
A) x>4 , no info about y so Insufficient
B) Y>-5 , no information about x so Insufficient

A+B, X>4 but Y can be negative or positive so as Y>-5( y= -5 or 0 or +4) so still Insufficient.

Hence answer is E
Thanks,

Hi lipsi18,

I hope you might want to consider the inference of Statement-1

There is no information about Y needed as you can see in the figure that when x co-ordinate is greater than 4 then the part of line falls in only 1st Quadrant where both x and y will essentially be positive hence x*y will be positive thereby giving the unique answer of the question.

Hence, SUFFICIENT

MANHATTAN GMAT OFFICIAL SOLUTION:

First, note that this is a Yes/No Data Sufficiency question.

Line L passes through three quadrants:

Quadrant I, where x and y are both positive, so xy > 0 and the answer is “yes.”
Quadrant III, where x and y are both negative, so xy > 0 and the answer is “yes.”
Quadrant IV, where x is positive and y is negative, so xy < 0 and the answer is “no.”

If you can determine what quadrant point P is in, you will have sufficient information to answer the question. Also, if you know that point P is in either quadrant I or quadrant III, that would also be sufficient.

(1) SUFFICIENT: If x > 4, then Point P is in Quadrant I, so xy > 0 and the answer is “yes.”

(2) INSUFFICIENT: If y > -5 , then Point P could be in either Quadrant I (xy > 0) or Quadrant IV (xy < 0).

The correct answer is A.
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Question: Is xy > 0?

For x*y > 0, the point should be either in 1st quadrant (both x and y positive) or in 3rd quadrant (both x and y negative)

Statement 1: x > 4

i.e. Point (x, y) is to the RIGHT of point S on line i.e. in 1st Quadrant, Hence

SUFFICIENT

Statement 2: y < -5

i.e. Point (x, y) is to the LEFT of point R on line i.e. in 3rd Quadrant, Hence

SUFFICIENT

Answer: Option D

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