M03-15

Question

If two lines have slopes  m  and  n  respectively, are they perpendicular?

(1)  m * n = -1 
 
(2)  m = - n

Bunuel · Accepted Answer

Official Solution:

If two lines have slopes  m  and  n  respectively, are they perpendicular? 
 
To determine if one line is perpendicular to another, we need to examine the relationship between their slopes. Two lines are perpendicular if their slopes are negative reciprocals of each other. This means if one line has a slope of  m , the line perpendicular to it should have a slope of  -\frac{1}{m} . Thus, the two lines will be perpendicular if and only if the product of their slopes is -1.
 
(1)  m*n = -1 . 
 
This directly answers the question with a YES.
 
(2)  m = -n . 
 
If  m=3  and  n=-3 , the lines are not perpendicular. However, if  m=1  and  n=-1  (or  m=-1  and  n=1 ), the product  m*n = -1 , confirming that the lines are perpendicular. Not sufficient.

Answer: A

Bunuel · Answer

The two lines are perpendicular if and only if the product of their slopes is -1.
 
If  m=3  and  n=-3 , then the product mn is not -1, so the lines are not perpendicular.
If  m=1  and  n=-1 , then the product mn is -1, so the lines are perpendicular.

Two different answers, hence not sufficient.

Prashant10692 · Answer

Hi Bunuel,
Could you please elaborate a little on Bold part.
(2) m=−nm=−n. Now,  if for example  m=3=−(−3)=−nm=3=−(−3)=−n then the lines are not perpendicular   but if m=1=−(−1)=−nm=1=−(−1)=−n,

So do you mean to say in the second case when we put the value in the reciprocal equation that is essential for perpendicular condition m*n=3*(-3)=-9 which is not the reciprocal i.e equal to -1. But in the case of 1 and -1 it is.

Bunuel · Answer

I've revised the question and solution, incorporating additional details for improved clarity. I trust this makes it more comprehensible.

M03-15

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