Two Line l and k Intersect at a Point (4, 3). Is the Product of their Slopes -1 GMAT Data Sufficiency

Question: Two line l and k intersect at a point (4, 3). Is the product of their slopes -1?

(1) x intercepts of line l and k are positive
(2) y intercept of line l and k are negative

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: B
Solution and Explanation:
Approach Solution 1:
IF....

X-intercept of K = 1
X-intercept of L = 2

The lines are NOT perpendicular and the answer to the question is NO.

IF.....

X-intercept of K = 1
X-intercept of L = 7

The lines ARE perpendicular and the answer to the question is YES.

Fact 1 is INSUFFICIENT
Fact 2: The Y-intercepts of both lines are negative.

This Fact proves that the slopes of BOTH lines are positive (since they cross the Y-axis at a negative number then move UP and to the RIGHT to the point (4,3)), so there is NO WAY for them to cross and be perpendicular. The answer to the question is ALWAYS NO.
Fact 2 is SUFFICIENT.

Approach Solution 2:
Statement 1:
x intercepts of line l and k are positive. Lines may be located on many options, including perpendicular manners. Answer Yes and No. So, INSUFFICIENT

Statement 2:
y intercepts of line l and k are negative. Both lines cannot have negative y intercept if they are perpendicular. Answer No. SUFFICIENT

Approach Solution 3:
If two lines are perpendicular to each other, the slope of one will be the negative reciprocal of the slope of the other (that is, if one slope is m, the other slope will be -1/m). 'Is the product of their slopes -1?' is just another way of asking whether this is true, since m(-1/m) = -1. So, translate the question as "are the two lines perpendicular?"

Now, let's draw. The goal is to prove insufficiency, so we'll be trying to draw, if possible, two pairs of lines for each statement: one pair that fits the statement and IS perpendicular, and one pair that fits the statement and ISN'T perpendicular. If that's impossible, then the statement must be sufficient.

Statement 1: both x intercepts are positive. That is, they both intercept the x axis to the right of 0.

Here's a perpendicular pair of lines that fits:

Here's a non perpendicular pair of lines that fits:

Statement 2: Both y intercepts are negative. That is, they both intercept the y axis below 0.
Here's a NON perpendicular pair of lines that fits:

However, it isn't possible to draw a pair of perpendicular lines that fit both the info in the question stem (intersect at (4,3)) and this statement. The angle between the two lines, marked here, is always going to be significantly less than 90 degrees if we only use lines that have negative y intercepts.

Therefore, using this statement, the answer to the question is definitely "no." So, the statement is sufficient and the answer is B.

“Two line l and k intersect at a point (4, 3). Is the product of their”- is a topic of the GMAT Quantitative reasoning section of GMAT. GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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