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 Post subject: GMAT Coordinate Geometry
PostPosted: Sat May 15, 2010 11:30 am 
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In the equation √(2 + m²) = m + 3, the variable m represents the slope of a line. What is the slope of a line that is perpendicular to the line whose slope is m?

A. -6/7
B. -7/6
C. 2/9
D. 6/7
E. 7/6

(D) First, we need to solve for m. Note, that m must be greater or equal to -3, because square root must be a non-negative number. If we square both sides of the equation, to eliminate the square root, we'll get:
2 + m² = m² + 6m + 9.
Then we proceed to solve for m:
2 + m² = m² + 6m + 9.
2 = 6m + 9
-7 = 6m
m = -7/6. -7/6 > -3, so it is solution of the equation.

Now we need to understand that the slopes of perpendicular lines are negative reciprocals of each other.

Therefore, all lines that are perpendicular to a line with a slope of -7/6 have slopes of 6/7.
The correct answer is choice (D).


Could you, please, explain why m is greater or equal to -3 and something regarding line slopes?


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 Post subject: Re: math: algebra & x,y-plane
PostPosted: Sat May 15, 2010 12:24 pm 
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When we solve an equation like √(2 + m²) = m + 3 and find a value (or values) of m, we need to make sure everything is Ok with original statement. For that we can either plug in a value (or values) of m in it:

√(2 + (-7/6)²) = (-7/6) + 3
√(2 + 49/36) = 11/6
√(121/36) = 11/6
11/6 = 11/6,

or we can set limitations for m right from the beginning:

√(2 + m²) = m + 3 applies following restrictions:
1. Since √(2 + m²) is a non-negative number then (m + 3) is also a non-negative number:
(m + 3) ≥ 0
m ≥ -3

2. Since square root of a number can be calculated only if the number is non-negative therefore 2 + m² ≥ 0
m² ≥ -2 is obviously true for any m.

Now, let me explain about the slope.
The equation of a line with slope (-7/6) is y = (-7/6)x + b. There are infinitely many lines that have slope -7/6:

Image

The equation of a line perpendicular to y = (-7/6)x + b is y = (+/7)x + c. There are also infinitely many lines that have slope 6/7:


Image

Since we are asked to find a slope, the answer is 6/7, choice (D).
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