Equidistant line to another line

[omer205]

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

Answer says that the equidistant line must pass in the middle of these two points...

I am somewhat not clear. I guess there might be multiple equidistant lines i.e. A line passing through origin and parallel to segment PQ will also be equidistant and will have the same slope as that of segment PQ...

Am I missing something ?

Thanks

[esledge]

You are not missing anything. There are two such lines, passing through the origin and equidistant from P and Q. One has a slope of 2.25 (passing between P and Q), the other a slope of -0.5 (parallel to PQ).

On Problem Solving questions with more than one correct answer, only one of the correct answers will be listed. Pick the one that is. However, the question would be more accurately phrased as "which of the following is the slope of a line that..." and the explanation should address both possibilities.

Can you please confirm exactly where you found this question (which book, edition, page, and question #)? I recall a similar fix to a similar problem, so this may have already been updated, but I'd like to make sure. Thanks.

[abhishekit]

I am somewhat not clear. I guess there might be multiple equidistant lines i.e. A line passing through origin and parallel to segment PQ will also be equidistant and will have the same slope as that of segment PQ...

Am I missing something ?

Thanks

A line parallel to PQ can never be equidistant from both P and Q. If you connect P and Q and draw a perpendicular bisector of PQ, ( a line that passes through the middle of PQ and is at right angles to PQ), this bisector will include ALL the possible points that are equidistant from both P and Q. Think about it this way, if a point does not lie on this bisector, it will be closer to either P or Q.
If the line passes through origin and is parallel to PQ, it means the distance between the two lines is constant. It DOES NOT mean that every point on the new line has the same distance from P and Q.

So it is clear, there can be only one such line and slope is given by the coordinates of the mid point.

[nimish.tiwari]

A line parallel to PQ can never be equidistant from both P and Q. If you connect P and Q and draw a perpendicular bisector of PQ, ( a line that passes through the middle of PQ and is at right angles to PQ), this bisector will include ALL the possible points that are equidistant from both P and Q. Think about it this way, if a point does not lie on this bisector, it will be closer to either P or Q.
If the line passes through origin and is parallel to PQ, it means the distance between the two lines is constant. It DOES NOT mean that every point on the new line has the same distance from P and Q.

So it is clear, there can be only one such line and slope is given by the coordinates of the mid point.

@abhishekit: my understanding of the question being asked is that its only to identify the "line" which is equidistant from both P & Q. The scenario you have mentioned, does fulfil this scenario and also contains all such points which are equidistant from P & Q. However, a line parallel to PQ would also be equidistant from P & Q (even if all the points on this line are not individually equidistant from P & Q). So this scenario is still correct in light of the question asked. However, as Ben has mentioned, its just a matter of finding which of the two scenarios' is mentioned in the options and go by that.

Cheers!!

[Ben Ku]

Please review responses from a previous thread on this question:
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