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Circle C and line K lie in the XY plane. If circle C is

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Circle C and line K lie in the XY plane. If circle C is [#permalink]

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Circle C and line K lie in the XY plane. If circle C is centered at the orgin and has a 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.

OPEN DISCUSSION OF THIS TOPIC IS HERE: http://gmatclub.com/forum/circle-c-and- ... 01471.html
[Reveal] Spoiler: OA

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Re: Circle C and line K lie in the XY plane. If circle C is [#permalink]

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New post 28 Apr 2010, 07:59
zz0vlb wrote:
Circle C and line K lie in the XY plane. If circle C is centered at the orgin and has a 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.


[Reveal] Spoiler:
E


Let the line is \(y -mx +c =0\)

Now if the distance from the center of circle > radius => Line lie outside circle and does intersect with the line.

Distance = \(\frac{|y1-mx1+c|}{\sqrt{(1+m^2)}}\)

Distance from (0,0) = \(\frac{|0*1-m*0+c|}{\sqrt{(1+m^2)}}\)
= \(\frac{|c|}{\sqrt{(1+m^2)}}\)

Since it depends upon both c and m , thus both the statements individually are not sufficient.
So rule out A B and D

Now take 2nd equation m = -1/10

Distance becomes =\(|c|/\sqrt{{1+\frac{1}{10}^2}}\)

d = \(|c|*\sqrt{\frac{100}{101}}\)

Now c>1, but still we cannot say anything about d as if c > \(\sqrt{\frac{101}{100}}\).
then d>1 else d<1 ( take the case when \(\sqrt{\frac{101}{100}}\) \(> c > 1\)

thus E
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Re: Circle C and line K lie in the XY plane. If circle C is [#permalink]

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New post 28 Apr 2010, 08:38
zz0vlb wrote:
Circle C and line K lie in the XY plane. If circle C is centered at the orgin and has a 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.

[Reveal] Spoiler:
E


No matter what the slope is, it’s possible for line not to cross the circle as the x intercept can be + infinite.

(1) Just says that x intercept is right to the circle. Not sufficient
(2) Just says that slope is -1/10 --> line is just going down. Not sufficient.

(1)+(2) As we don't know exact intercept of line and X-axis we can not determine whether line intersects the circle or not. Not sufficient.

To elaborate more: we can draw infinitely many parallel lines with X-intercept more than 1 and slope -1/10, some will intersect the circle and some not.

Answer: E.
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Re: Circle C and line K lie in the XY plane. If circle C is [#permalink]

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New post 22 Aug 2010, 18:53
I understand this one a little bit but I'd like some more clarity. As far as statement 2 goes, other than the fact that the slope is negative, what else can I gather from it. Thanks

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

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Re: Circle C and line K lie in the XY plane. If circle C is [#permalink]

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New post 31 Aug 2010, 23:03
Good explanation by the master!!

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Re: Circle C and line K lie in the XY plane. If circle C is [#permalink]

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New post 01 Sep 2010, 09:41
+1 E

The line may be under the circle and very far. It will never touch the circle.

I think I deserve kudos ;)
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Re: Circle C and line K lie in the XY plane. If circle C is [#permalink]

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New post 22 Mar 2011, 17:50
Circle C and line k lie in the xy plane. If circle C is centered at the origin and has radius 1, does link k intersect circle C?

1) X-intercept of line k is greater than 1
2) The slope of line k is -1/10

Can somebody explain for me?

And, I generally get scared of co-ordinate geometry, Can somebody point me to the right material?

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Re: Circle C and line K lie in the XY plane. If circle C is [#permalink]

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nvuppala wrote:
Circle C and line k lie in the xy plane. If circle C is centered at the origin and has radius 1, does link k intersect circle C?

1) X-intercept of line k is greater than 1
2) The slope of line k is -1/10

Can somebody explain for me?

And, I generally get scared of co-ordinate geometry, Can somebody point me to the right material?


Let the equation of the line be \(y=mx+c\)where m is the slope and c is the y intercept. Then, the x intercept of this line is \(-c/m\)

Now, a line would intersect the given circle only if the length of perpendicular from its centre on the given line is less than or equal to the radius.

Length of perpendicular from centre (origin in this case) to the given line is given by \(|c/\sqrt{(1+m^2)}|\) . This comes from the formula for the perpendicular distance of a point from a line.

We need to find if \(|c/\sqrt{(1+m^2)}|\) is less than or equal to 1 (as radius is 1 in this case).

Statement 1 gives -c/m is greater than 1. or c is less than -m. But under given condition \(|c/\sqrt{(1+m^2)}|\) can be greater than or less than one for various values of c and m. So, insufficient

Statement 2 gives, m = -1/10. Here again, \(|c/\sqrt{(1+m^2)}|\) can be less than or more than 1 for various values of c. So, insufficient

Combining 1) and 2) also, we cant ascertain for sure if \(|c/\sqrt{(1+m^2)}|\) would always be less than or equal to 1. So, insufficient.

Hence, Answer is E.

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Re: Circle C and line K lie in the XY plane. If circle C is [#permalink]

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New post 23 Mar 2011, 21:59
I believe this fancy question is equivalent to CR "evaluate". The critical question can be rephrased as can you determine the equation of the line?With the information in both s1 and s2 it can't be done. So the answer is E. Pls verify the reasoning.

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Re: Circle C and line K lie in the XY plane. If circle C is [#permalink]

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New post 23 Mar 2011, 22:16
gmat1220 wrote:
I believe this fancy question is equivalent to CR "evaluate". The critical question can be rephrased as can you determine the equation of the line?With the information in both s1 and s2 it can't be done. So the answer is E. Pls verify the reasoning.

Posted from my mobile device


Well - the reasoning is not correct. For determining the equation of the line, we need definite values of c and m. However, in current case, we just need to know whether the value \(|c/\sqrt{(1+m^2)}|\) is less than or equal to 1. Statements could have told us something like c is greater than 10 and m lies between 0 and 1. In such a case, we cant determine the equation of line but we can certainly know that \(|c/\sqrt{(1+m^2)}|\) is greater than 1.

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Re: Circle C and line K lie in the XY plane. If circle C is [#permalink]

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New post 23 Mar 2011, 22:43
Let's say c is greater than 10. Worst case c is near 10. The angle that line makes with x axis is between 0 and 45 deg. I can draw the lines and be sure if it touches the circle. This is suff to answer the question. :-) but here I don't see the question is providing me enough info to fancy anything about the line itself.

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Re: Circle C and line K lie in the XY plane. If circle C is [#permalink]

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gmat1220 wrote:
Let's say c is greater than 10. Worst case c is near 10. The angle that line makes with x axis is between 0 and 45 deg. I can draw the lines and be sure if it touches the circle. This is suff to answer the question. :-) but here I don't see the question is providing me enough info to fancy anything about the line itself.

Posted from my mobile device

The point I was making was that you DONT need the equation of line to answer this question, just a condition on c and m can help us answer the question and hence your reasoning in the earlier post that question can rephrased as "Can we determine the equation of the line" is not correct. Hope this clarifies.

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Re: Circle C and line K lie in the XY plane. If circle C is [#permalink]

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New post 24 Mar 2011, 20:05
I totally agree beyondgmatscore. Thanks

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Re: Circle C and line K lie in the XY plane. If circle C is [#permalink]

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New post 14 Aug 2017, 07:46
Circle C and line K lie in the XY plane. If circle C is centered at the origin and has a radius 1, does line K intersect circle C?

The best way to solve this question would be to visualize/draw it.

No matter what the slope is, it’s possible for line not to cross the circle as the x intercept can be + infinite.

(1) The X-Intercept of line k is greater than 1 --> Just says that X-intercept is to the right of the circle. Not sufficient
(2) The slope of line k is -1/10 --> Just says that slope is negative -1/10 --> line is just going down. Not sufficient.

(1)+(2) As we don't know exact intercept of line and X-axis we can not determine whether line intersects the circle or not. Not sufficient.

To elaborate more: we can draw infinitely many parallel lines with X-intercept more than 1 and slope -1/10, some will intersect the circle (for example line with X-intercept 1.1) and some not (for example line with X-intercept 1,000,000). Check the image below for two possible scenarios: blue line (with the slope of -1/10 and the x-Intercept greater than 1) intersects the circle while the red line (also with the slope of -1/10 and the x-Intercept greater than 1) does not.

Image

Answer: E.

For more on this issue check Coordinate Geometry Chapter of Math Book: http://gmatclub.com/forum/math-coordina ... 87652.html

OPEN DISCUSSION OF THIS TOPIC IS HERE: http://gmatclub.com/forum/circle-c-and- ... 01471.html
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Re: Circle C and line K lie in the XY plane. If circle C is   [#permalink] 14 Aug 2017, 07:46
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Circle C and line K lie in the XY plane. If circle C is

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