Data Sufficiency | Coordinate Geometry : Quadrants

Data Sufficiency in Coordinate Geometry.

Here is an interesting GMAT data sufficiency question on intercepts and points of intersection of two perpendicular lines. A medium difficulty GMAT 650 to 700 level sample question.

Data Sufficiency Question

Line L is perpendicular to line K whose equation is 3y = 4x + 12; Lines L and K intersect at (p, q).
Is p + q > 0?

1. x intercept of Line L is less that that of Line K
2. y intercept of Line L is less than that of Line K

Correct Answer : Choice A. Statement 1 alone is sufficient to answer the question; statement 2 is not sufficient.

Explanatory Answer | Data Sufficiency

Equation of line K is 3y = 4x + 12. It is a positive sloping line intercepting the x axis at (-3, 0) and the y axis at (0, 4).

Line L is perpendicular to line K. The coordinates of the point of intersection of lines L and K are (p, q).

If (p, q) is in the first quadrant, then p + q > 0. If (p, q) is in the third quadrant, then p + q < 0. The third possibility is that (p, q) is in the second quadrant. If it is in the second quadrant, we will need specific inputs about the point of intersection to determine whether p + q is positive.

Evaluate Statement 1 Alone

Statement 1 – x intercept of Line L is less that that of Line K

The red line represents the scenario where lines L and K, both have the same x – intercept i.e., (-3, 0)

However, as per statement 1, the x intercept of line L is less than that of line K.
So, the dotted line represents line L. If the x intercept of line L is less than that of line K, the point of intersection of the two lines will be further to the left of (-3, 0).

It is quite evident from the diagram that the two lines will therefore, intersect in the 3rd quadrant.

Any point in the 3rd quadrant will have negative x and negative y values.

Therefore, p + q < 0.
Statement 1 is SUFFICIENT.

Evaluate Statement 2 Alone

Statement 2 – y intercept of Line L is less than that of Line K

In the above diagram two variants for line L are presented.

Option 1: In the variant where line L is in red, the y intercept of L is less than that of line K at (0, a). It can be seen that the point of intersection of line L (red) and line K (black line) is in the second quadrant. In the 2nd quadrant x coordinate is negative and y coordinate is positive.

Therefore, p + q could be positive or negative or 0. Cannot determine.

Option 2: In the variant where line L is represented in blue, the y intercept of L is less than that of line K at (0, b). Therefore, it can be seen that the point of intersection of line L (blue) and line K (black) is in the 3rd quadrant. Hence, in option 2, p + q < 0.

Because we are not sure from statement 2 which of the options line L could be, we cannot get a definite answer to the question.

Statement 2 is INSUFFICIENT.

Statement 1 alone is sufficient; statement 2 alone is not sufficient.

Correct answer Choice A.