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## Coordinate Geometry

tagged by: Brent@GMATPrepNow

shibsriz@gmail.com Master | Next Rank: 500 Posts
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#### Coordinate Geometry

Fri Jul 11, 2014 5:55 am
Hello Experts,

Pls explain this one:
Attachments

### GMAT/MBA Expert

Brent@GMATPrepNow GMAT Instructor
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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

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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Last edited by Brent@GMATPrepNow on Mon Apr 16, 2018 1:00 pm; edited 1 time in total

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### Top Member

GMATinsight Legendary Member
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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

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