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?

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:

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:

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:

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

Users browsing this forum: No registered users and 0 guests

You cannot post new topics in this forum You cannot reply to topics in this forum You cannot edit your posts in this forum You cannot delete your posts in this forum You cannot post attachments in this forum

GMAT(TM) and GMAT CAT (TM) are registered trademarks of the Graduate Management Admission Council(TM). The Graduate Management Admission Council(TM) does not endorse, nor is affiliated in any way with the owner or any content of this site.