MANHATTAN GMAT CAT 1
In the xy-coordinate system, what is the slope of the line that goes through the origin and is equidistant from the two points P = (1, 11) and Q = (7, 7)?
a) 2
b) 2.25
c) 2.50
d) 2.75
e) 3
The question asks us to find the slope of the line that goes through the origin and is equidistant from the two points P=(1, 11) and Q=(7, 7). It's given that the origin is one point on the requested line, so if we can find another point known to be on the line we can calculate its slope. Incredibly the midpoint of the line segment between P and Q is also on the requested line, so all we have to do is calculate the midpoint between P and Q! (This proof is given below).

Let's call R the midpoint of the line segment between P and Q. R's coordinates will just be the respective average of P's and Q's coordinates. Therefore R's x-coordinate equals 4 , the average of 1 and 7. Its y-coordinate equals 9, the average of 11 and 7. So R=(4, 9).

Finally, the slope from the (0, 0) to (4, 9) equals 9/4, which equals 2.25 in decimal form.

My thinking
" It is not necessary that line has to pass through the mid point of two coordinates. It can be parallel to the line passing through these two points and origin"

Any line that is parallel to another line will have the same slope - so, yes, that's fine. If you've been given specific info to answer a question, though, then the easiest approach is to use that given info. The easiest way to find two points on this line: use the given point (the origin) and calculate the midpoint of the two points the line lies equidistant between.

I am not quite sure about the truth of the andwer to the problem. Because when we talk about the distance of one point to the line, we talk about the length of line segment, from point to the line, that is perpendicular to the line. The question only says that this perpendicular line segments are equal in length. Nothing more. So the line need not be in the middle.

This problem has been discussed in another thread. Your point sounds similar to the ones raise there, so check it out. FYI, we have the problem under review internally. Thanks!

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