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Coordinate Geometry

This topic has 1 expert reply and 1 member reply
shibsriz@gmail.com Master | Next Rank: 500 Posts Default Avatar
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Coordinate Geometry

Post Fri Jul 11, 2014 5:55 am
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    Hello Experts,


    Pls explain this one:
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    Post Fri Jul 11, 2014 6:42 am
    Note: I've reworded the target question so that it's an actual question.

    Quote:
    Does the line y1 = mx + b pass through the point (2, -1)?

    1) y1 is perpendicular to the line y2 = -(1/2)x + 9
    2) b = -5
    IMPORTANT: For geometry (including coordinate geometry) Data Sufficiency questions, we are typically checking to see whether the statements "lock" a particular line, angle, length, or shape into having just one possible measurement. This concept is discussed in much greater detail in our free video: http://www.gmatprepnow.com/module/gmat-data-sufficiency?id=1103

    This technique can save a lot of time.

    Here, we want to determine whether or not the statements lock us into having 1 and only 1 line.

    Target question: Does the line y1 = mx + b pass through the point (2, -1)?

    Statement 1: y1 is perpendicular to the line y2 = -(1/2)x + 9
    Here's the line y2 = -(1/2)x + 9


    As you can see, there are MANY MANY MANY lines that are perpendicular to this line.


    So, the line y1 = mx + b MAY or MAY NOT pass through the point (2, -1))
    Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

    Statement 2: b = -5
    This tells us that the line y1 = mx + b has y-intercept -5
    In other words, the line y1 = mx + b must pass through the point (0, -5)


    There are MANY MANY MANY different lines that pass through the point (0, -5)


    As you can see, the line y1 = mx + b MAY or MAY NOT pass through the point (2, -1))
    Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

    Statements 1 and 2 combined
    Statement 1 tells us that the line y1 = mx + b can be among the many dotted red lines that are perpendicular to the green line.

    Statement 2 says that the line must also pass through the point (0, -5)
    This LOCKS us into 1 and only 1 line.

    Since the statements lock us into just 1 line, we have enough information to determine whether or not
    the line y1 = mx + b passes through the point (2, -1))
    Since we can answer the target question with certainty, the combined statements are SUFFICIENT

    Answer = C

    IMPORTANT: Need we find the equation of the line? Need we actually determine whether or not the line passes through (2, -1)? No and no. We need only recognize that, since there's only 1 line that satisfies both statements, we COULD find the equation of the line and we COULD determine whether is passes through (2, -1)

    Cheers,
    Brent

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    GMATinsight Legendary Member
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    Post Fri Jul 11, 2014 6:58 am
    For (2,-1) to lie on line y1 = mx+b when you substitute the value of x=2 then y should be obtained as -1 [Point (2,-1) should satisfy the equation of line given]

    Statement 1) Line y2 is perpendicular to Line y1

    This leads to the Product of their slopes should be -1
    i.e. (-1/2)x m = -1
    i.e. m = 2

    Thefore the equation of line y1 become y1 = 2x+b
    But since b is unknown therefore insufficient to answer the question

    Statement 2) b=-5
    Doesn't give us any information to calculate slope of line therefore Insufficient.

    Combining the two statements
    the equation of line becomes
    y1 = 2x-5
    Now we can substitute the values (2,-1) and check whether the equation is satisfied or not

    SUFFICIENT

    Answer: Option C

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