In the xy-Plane, Line k Passes Through The Point (1, 1) And Line m GMAT Data Sufficiency

Question: In the xy-plane, line k passes through the point (1, 1) and line m passes through the point (1, -1). Are the lines k and m perpendicular to each other ?
(1) Lines k and m intersect at the point (1, -1)
(2) Line k intersects the x-axis at the point (1, 0)

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.

‘In the xy-plane, line k passes through the point (1, 1) and line m passes through the point (1, -1)’ - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". 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.

Solution and Explanation

Approach Solution 1:

There is only one approach to solve the question.
For one line to be perpendicular to another, their slopes must be negative reciprocals of each other (if slope of one line is m than the slope of the line perpendicular to this line is
−1/m).
In other words, the two lines are perpendicular if and only the product of their slopes is
−1.
So basically the question is can we somehow calculate the slopes of these lines.
From stem we have one point for each line.
(1) gives us the second point of line k, hence we can get the slope of this line, but we still know only one point of line m.

• The statement is Not sufficient.

(2) again gives the second point of line k, hence we can get the slope of this line, but we still know only one point of line m.

• The statement is Not sufficient.

(1)+(2)
we can derive the slope of line k but for line m we still have only one point, hence we can not calculate its slope.

• The statement is Not sufficient.

None of them are sufficient and hence, E is the correct answer.

Correct Answer: E

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