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How to do this geometry problem..?

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How to do this geometry problem..?

by praveen_gmat » Tue Aug 17, 2010 8:05 am
Circle C and line k lie in the xy-plane. If circle C is centered at the origin and has radius 1, does line k intersect
circle C?
(1) The x-intercept of line k is greater than 1 (2) the slope of line k is -1/10

-----

I figures out that the equation of the line should y=-x+1 ... but then could not proceed.
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by skprocks » Tue Aug 17, 2010 9:07 am
praveen_gmat wrote:
Circle C and line k lie in the xy-plane. If circle C is centered at the origin and has radius 1, does line k intersect circle C?
(1) The x-intercept of line k is greater than 1
(2) the slope of line k is -1/10

-----

I figures out that the equation of the line should y=-x+1 ... but then could not proceed.
I could not understand,How is the eqn representing the line is defying both the statements 1 as well as 2 ?
According to the eqn you have mentioned, y=-x+1

Slope = -1 (Line's eqn of form y=mx+C)
And,It is against both (1) and (2) as mentioned in the problem.

IMO Let the Line k be represented by the eqn. y=mx+c
From (1)

-c/m > 1
INSUFF alone

and From(2)

m=-1/10
INSUFF alone

Combining (1) and (2)

-c/m > 1
i.e. -c > m
i.e. -c> -1/10
multiplying by -1 on both sides of inequality,inverts the inequality symbol

c < 1/10

Thus, c can be ignored.(My assumption,Please correct me in case its Wrong)

Circles eqn : x^2+y^2=1
Line : y= -1/10X + 0.01 (assuming c=0.01)

Answer: [spoiler](E)[/spoiler]
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by uwhusky » Tue Aug 17, 2010 9:10 am
Is the answer C?

I approached this question by drawing the picture. Circle C centered at the origin with radius of 1 means that it is touching the following coordinates: (1, 0), (0, 1), (-1, 0), (0, -1).

1) The x-intercept of line k is greater than 1. Assuming that the x-intercept is 2, it means that at least one point of line k is on (0, 2). Insufficient, we need slope to know if line k intersects the C.

2) the slope of line k is -1/10, insufficient, because we don't know where the x-intercept is.

Together sufficient, because we know that if x-intercept is above 1 and the slope is less than 1, line k won't intersect C.

Please let me know if x-intercept could be like 1.00000001 and the slope of -1/10 allows line k to "scrape" a tiny piece of C...then I guess the answer would be E, but I would hate whoever wrote this question.
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by CosmicEnergy » Tue Aug 17, 2010 9:55 am
praveen_gmat wrote:Circle C and line k lie in the xy-plane. If circle C is centered at the origin and has radius 1, does line k intersect
circle C?
(1) The x-intercept of line k is greater than 1 (2) the slope of line k is -1/10

-----

I figures out that the equation of the line should y=-x+1 ... but then could not proceed.
Well, we do not really need to find the equation of the line.
These questions are best solved without solving :-)

1) realize that we can draw a line with x intercept greater than 1 by cutting the circle as well as by not cutting the circle
NOT SUFFICIENT

2) Slope is -1/10
As slope is negative, the line has to go down from left to right; here also we can draw the line by cutting the circle as well as by not cutting the circle.
NOT SUFFICIENT

Together
slope is negative and x-intercept >1

Draw and see yourself again that you can achieve this either ways

Ans E

NOTE: whenever you see intercept question, it is a test of possibilities (permutations) :-)
GMAT loves to test this concept often; make sure you understand the concept of slope: https://www.mathopenref.com/coordslope.html

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by uwhusky » Tue Aug 17, 2010 10:33 am
CosmicEnergy,

Can you see where my assumption went wrong? How do you conclude that line k could both intercept and not intercept C, especially knowing that this is a DS question and the more obvious choice being E for this question.
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by CosmicEnergy » Tue Aug 17, 2010 7:30 pm
uwhusky wrote:CosmicEnergy,

Can you see where my assumption went wrong? How do you conclude that line k could both intercept and not intercept C, especially knowing that this is a DS question and the more obvious choice being E for this question.
Dear uwhusky

I guess, you assumed that the line is cutting the x-axis near the circle;
consider this: Take a point at x= 10,000 and draw a line with -1/10 slope; it will definitely not cut the circle.

HTH.
Soot back should you have further question.

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by uwhusky » Tue Aug 17, 2010 7:35 pm
Actually I am assuming that the line will never cut the circle, thus C. I am looking for explanations on how the line will actually cut into the circle resulting in E.
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by CosmicEnergy » Tue Aug 17, 2010 8:24 pm
uwhusky wrote:Actually I am assuming that the line will never cut the circle, thus C. I am looking for explanations on how the line will actually cut into the circle resulting in E.
When x= 1.001; it has to cut the circle because the slope -1/10 will make a very very small angle (clockwise) with the x axis.

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by uwhusky » Tue Aug 17, 2010 9:15 pm
Will GMAT ever ask a question like this one? It's really not testing anything...
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by CosmicEnergy » Wed Aug 18, 2010 1:38 am
uwhusky wrote:Will GMAT ever ask a question like this one? It's really not testing anything...
Dear uwhusky

This question is of GMAT quality for sure.
When you are scoring above 700, you will get concept oriented questions.
For example, in SC, it is likely that, if you are in 700+ range, you will get questions of meaning clarity rather than of grammar .


Anyway, what is the source of this question?

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by uwhusky » Wed Aug 18, 2010 2:49 am
I am dumb...x-intercept crosses x-axis, y-intercept crosses y-axis. Shoosh, I was thinking of y-axis > 1, which means that the line would be in quadrant I and II...

I guess it is still able to intersect the circle, but very tiny bit of it, and in which case, I doubt GMAT would test that.
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