Data Sufficiency

Q: If a line is tangent to circle, center at the origin. What is the slope of line ?

A: Line is tangent to circle at the X- axis at point ( 2, 0)
B: line pass through ( 3, 4)

IMO C

cans wrote:IMO C

please explain the method/answer

Is the answer correct?

[email protected] wrote:Q: If a line is tangent to circle, center at the origin. What is the slope of line ?

A: Line is tangent to circle at the X- axis at point ( 2, 0)
B: line pass through ( 3, 4)

the answer is eeeee

Interesting question with a few pretty key takeaways.

I'd say that the answer is A* but let met put an asterisk next to that on a technicality that I don't think could show up on the GMAT.

Note that the tangent line to a circle runs perpendicular to the diameter of the circle that also goes through the same point. So the tangent running through (2,0) must be perpendicular to the diameter there, and if the center of the circle is (0,0), then that diameter is a horizontal line...it's the x axis. So we do know exactly the tangent line here - it's x = 2.

Now, even if you were only familiar with that rule of tangent lines but weren't 100% sure, here's a pretty big clue to you...C is a little too obvious an answer here. With both statements together you're given two points on the same line, which makes for a quick calculation for the slope. So since C is a little too easy of an answer, you want to consider whether one of the statements alone will be sufficient on its own, and you can reinvestigate statement 1. Not only does 1 give you a point, but it adds that definition that it's not just "any" point, it's the tangent point. That should make you strongly consider choice A, and if were even to draw a few circles and lines you'd probably be able to see that a tangent line at a single point has to have a defined slope there...you can't draw two different lines that will be tangent at the exact same point.

Again, strategically - because answer choice C is just a little too easy, that's why you'd do that extra work on statement 1 and I'd even suggest guessing A instead of C if you didn't know for sure that tangent rule but had a pretty good inclination that there is a rule. That's DS strategy - don't accept an "obvious" answer without further investigating everything you know about the statements.

Now for the asterisk...because the line is x = 2, technically the slope is "undefined" and so A is kind of a tricky answer because you don't get a number. But I'd be shocked if the GMAT used "undefined" as a "gotcha" trap on this one...it's just not part of their reward system, and since the slope is only "undefined" for one type of line (a perfectly vertical line) that's a weird technicality that I honestly don't know how they'd define and I don't think they could ever use it for that reason - too many people who *should* get this problem right (the 700+ crowd) would get it wrong by guessing the wrong way (is "undefined" sufficient if there's only one "undefined" slope?).

Note, also, that we can prove that this isn't an official problem because statement 2 contradicts statement 1 (point 3,4 isn't on the line x = 2), so I wouldn't worry about that asterisk/technicality here. The lesson to be learned, though, is pretty beneficial.

Brian@VeritasPrep wrote:Interesting question with a few pretty key takeaways.

I'd say that the answer is A* but let met put an asterisk next to that on a technicality that I don't think could show up on the GMAT.

Note that the tangent line to a circle runs perpendicular to the diameter of the circle that also goes through the same point. So the tangent running through (2,0) must be perpendicular to the diameter there, and if the center of the circle is (0,0), then that diameter is a horizontal line...it's the x axis. So we do know exactly the tangent line here - it's x = 2.

Now, even if you were only familiar with that rule of tangent lines but weren't 100% sure, here's a pretty big clue to you...C is a little too obvious an answer here. With both statements together you're given two points on the same line, which makes for a quick calculation for the slope. So since C is a little too easy of an answer, you want to consider whether one of the statements alone will be sufficient on its own, and you can reinvestigate statement 1. Not only does 1 give you a point, but it adds that definition that it's not just "any" point, it's the tangent point. That should make you strongly consider choice A, and if were even to draw a few circles and lines you'd probably be able to see that a tangent line at a single point has to have a defined slope there...you can't draw two different lines that will be tangent at the exact same point.

Again, strategically - because answer choice C is just a little too easy, that's why you'd do that extra work on statement 1 and I'd even suggest guessing A instead of C if you didn't know for sure that tangent rule but had a pretty good inclination that there is a rule. That's DS strategy - don't accept an "obvious" answer without further investigating everything you know about the statements.

Now for the asterisk...because the line is x = 2, technically the slope is "undefined" and so A is kind of a tricky answer because you don't get a number. But I'd be shocked if the GMAT used "undefined" as a "gotcha" trap on this one...it's just not part of their reward system, and since the slope is only "undefined" for one type of line (a perfectly vertical line) that's a weird technicality that I honestly don't know how they'd define and I don't think they could ever use it for that reason - too many people who *should* get this problem right (the 700+ crowd) would get it wrong by guessing the wrong way (is "undefined" sufficient if there's only one "undefined" slope?).

Note, also, that we can prove that this isn't an official problem because statement 2 contradicts statement 1 (point 3,4 isn't on the line x = 2), so I wouldn't worry about that asterisk/technicality here. The lesson to be learned, though, is pretty beneficial.

THANKS BRAIN. I LIKE YOUR EXPLANATION.
BUT I THINK THE ANSWER IS EEEEEEEE. THE LINE CAN PASS THROUGH 2 AND 4 QUADRANT AND CAN BE TANGENT TO CIRCLE AT (2, 0). IN THIS CASE THE SLOPE IS NEGATIVE.
BUT IT CAN ALSO PASS THROUGH 1 AND 3 QUADRANT AND CAN BE TANGENT TO CIRCLE. THUS IN THIS CASE THE SLOPE IS NEGATIVE.

PLEASE HELP REGARDING THIS QUESTION

Hmmm...I think you're probably misreading the question. The line is tangent to the circle. The circle has a center on the origin. Which means that the line CANNOT go through the origin because "tangent" means that it touches the circle exactly once - it does not cut through the circle. So the line cannot go through quadrants 1 and 3 only or through quadrants 2 and 4 only.

We also know that the tangent must be perpendicular to the diameter at that point, so we know exactly the angle of the line. The diameter is along the x-axis, which means that the perpendicular tangent will parallel the y-axis. If you want a quick visual, check out https://www.mathopenref.com/tangent.html (note - no real endorsement of that exact site...it's just a decent visual of that rule).

Correct me if i am wrong.
line is tangent at (2,0) means it passes through (2,0)
also b) given another point through which line passes.
So combining them, we get 2 points which make only 1 line. and thus we can get the slope.
So c) sufficient.

if it were not for tangent to cross the circle in the only point (2;0) then yes it would be C, but it was tangent and st(1)&st(2) would result in contradictions and undefined slope with A as Brian explained.

cans wrote:Correct me if i am wrong.
line is tangent at (2,0) means it passes through (2,0)
also b) given another point through which line passes.
So combining them, we get 2 points which make only 1 line. and thus we can get the slope.
So c) sufficient.

The prompt also says that the circle is centered at the origin, does that not give us sufficient information to find out the slope with both statement 1 and 2?

We know that in statement 1 the x intercept is (2,0). slope = undefined as the tangent is perpendicular to the x axis

Brian, when evaluating statement 2 why should I look at statement 1?

Thanks, pemdas - and, cans, to add to that... the problem with C isn't the "both statements together are sufficient" portion, it's the "but NEITHER ALONE" portion. Yes, knowing two points on the same line will give you the slope, but actually statement 1 ALONE is sufficient. The fact that the circle is centered on (0,0) and that our line is tangent to that circle at (2,0) means that our line runs perpendicular to the x-axis. We can now draw that line...it's x = 2. We don't need any more information, so we don't need to bring in statement 2 (to get C) - we already have all the information we need with statement 1 alone.

And, sl750 - that's why I bring up that you can play one statement off of the other. It's a step above in terms of strategy...think of it this way:

1) When assessing statement 2, DO NOT include the information you learned from statement 1. Remember, choice B is "Statement 2 ALONE"...you need to have a full read on statement 2 independent of any other information.

BUT

2) Statement 2 doesn't exist in a vacuum. It's part of a larger question, which the author wrote in very carefully-chosen terms. Therefore, statement 1 can provide you with some clues that will be helpful in independently assessing statement 2. Consider, for example, the question:

What is the value of x?

(1) x is positive

(2) x^2 = 25

Hopefully this is a pretty basic example, but pay attention to the lesson. You DO NOT want to assess statement 2 with the idea that x is positive. You'd get this question wrong if you did that...while looking at statement 2 we only know that x^2 = 25. But say you were to take the square root of 25 and say that x = 5. Now you really should look back at statement 1, which says that x is positive. Do you need that information? Or does statement 2 on its own guarantee that information? Had you forgotten about the potential for x to be -5 (based on x^2 = 25), statement 1 can remind you that you need to explicitly make that decision... "Do I need an extra statement to tell me that x is positive?". That should encourage you to ask "based on statement 2, could x be negative?". And it can, so having looked at statement 1 before making a final determination on statement 2, you can save yourself from a bad answer.

_______________________________________________

On this coordinate question, I think that looking at statement 2 before you finalize your answer on statement 1 can be instrumental in getting the right answer. If all you take from statement 1 is that "the line goes through (2,0)", then you'd say "Not Sufficient". But looking at statement 2, which gives you a second point and is similarly not sufficient, that should give you pause. C is a gift answer here...two statements each give you one point so together you can take (y1 - y2)/(x1 - x2) and you have your slope, quickly. That should raise a flag in your mind - why was C so obvious? What am I missing? And since statement 1 has more information ("tangent at point 2,0") than statement 2 ("runs through point 3,4"), then you most likely want to double-check statement 1 through the lens of "wait, do I really need them to tell me statement 2?".

I think of it like a chess match - there are two dimensions here when you see an opponent pick up a piece, move it to a square, ponder the move, then take it back. You're not at all "wrong" if you ignore that move and just play based on where the pieces are on the board. But you're a better player and more likely to win if you think about what the opponent was setting up by thinking about that move. Even though that move doesn't exist officially, you know that it's in your opponent's move...he may be trying to defend his queen or hoping to attack your back line. Knowing what's on his mind can help you think a few moves ahead.

Well, considering the POTENTIAL (but not actual) impact of "the other" statement in DS is the same thing. Yeah, you don't know statement 2 yet when you're reading statement 1. But the author of the question wrote both and the question format allows you to take a sneak peak at statement 2 before you make your final decision on statement 1, so why not give it a look to see if you can get inside the head of the author?