Is (-1,-1) a point on line OQ?

P must be in the second quadrant and Q must be in the third quadrant. But you can rotate within them. Consider the diagram below:So, as you can see, the second statement alone is not sufficient.

Hope it's clear.

(1)+(2) Since P lies on line y=x and angle POQ = 90 degrees, then point Q (line segment OQ) lies on line y=x. Therefore, point (-1,-1) lies on line OQ. Sufficient.

hello Bunuel,

I don't understand the above explanation when you combine the two statements, P Lies on y =-x and angle POQ =90 then how do we conclude that point Q lies on y=x. Do P and Q have the value of x coordinate ? Is that info provided from figure?

Lines y=x and y=-x are perpendicular to each other and both pass through the origin:OP is part of y=-x (notice that O is the origin), thus OQ must be part of y=x.

(1) Nothing about the co-ordinates Q can be stated with certainty.
(2) This tells that angle POQ is 90 deg. Nothing concrete about Q.

(1) + (2) If a line joining P and Q is drawn, then a right angled triangle POQ forms. OP can be calculated. Assuming Q lies on the line OQ, distance OQ can be calculated. Then the hypotenuse can calculated. If it satisfies Pythagorean theorem, then (-1,-1) lies on OQ, else it does not. Hence, (C). Note that there is no need to compute any of the lengths. We just need to think whether the the lengths can be computed. If they can (which is the case here), then (-1,-1) may or may not be a point on OQ, hence (C).

Statement1) Gives us slope of the line passing through PQ as -1. Says nothing about line OQ
Insufficient

Statement 2) tells us that slope of line PQ will be a inverse reciprocal of the line OQ, but does not give any info about slope
Insufficient

Merge both statements

PQ's slope = -1
OQ slope = 1
Hence OQ will definitely pass from (-1,-1)

Hi Bunnel, In case the angle POQ was given to be anything other than 90 degrees then what would be the answer?

The answer would still be C but the answer to the question would be NO, point (-1,-1) doe not lie on line OQ.

Hi Bunnel, Could you please explain as to how we will be able to tell that the point will not lie on OQ if the angle was say 60 degrees. Thanks

No matter what is the measure of angle POQ, line segment OQ would be fixed (because P is fixed the measure of angle POQ will fix the line segment OQ), so we would be able to answer whether it contains point (-1,-1) . Also, the only way it to contain (-1,-1) is when OQ is on y=x, or in other words when POQ is 90 degrees.

Bunuel - Please clear my thought process as when we combine the statements it states that P=(-2,-2) and angle POQ is 90 deg

1) Can we assume from this and the figure that the two points are figuratively on the opposite ends as they lie between y=x and y=-x and can jump on the conclusion that OQ does carry the point (-1,-1).

2) Secondly not able to figure out the meaning of the rotation within them?

Thanks in advance.

When we combine the statements we can infer that P is in the II quadrant and somewhere on line y = -x. Similarly we can infer Q is in the III quadrant and somewhere on line y = x.

Statement 1: Only tells the location of point P. Insufficient
Statement 2: Only tells that both lines are perpendicular to each other at origin (0,0)

Combining statement 1&2: we know that lines are perpendicular to each other so the slope of one line will be the negative reciprocal of the other line. Using point P(-2,2) and O(0,0), the slope of line OP is -1. so the slope of line OQ will be the negative reciprocal of -1 i.e. 1.

If the slope, using points (-1,-1) and (0,0), is also 1, then it means point (-1,-1) lies on line OQ.

using the slope formula and using points (-1,-1) and (0,0): m = (-1-0)/(-1-0) = 1

so point (-1,-1) lies on line OQ.

Answer: C

Just wanted to know that those small lines on line segment op and oq don't mean that they equal?

Those lines meant to be dots I think but whether OP = OQ is actually irrelevant.

If they are equal, won't it mean that oq is (-2,-2)?

You mean from (1)? No, if we were told that OP = OQ and P = (-2,2), then it would not necessarily mean that Q = (-2, -2). We don't know the measure of POQ - if it's 90 degrees then yes, Q = (-2, -2) but if not, then Q will not be at (-2, -2). But again the lengths of OP and OQ are absolutely irrelevant here. We re asked to find whether (-1,-1) is a point on line OQ, so we need to know the slope of line OQ, not the length of line segment OQ.

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