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Circle C and line k lie in the xy plane. If center of C lie

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Circle C and line k lie in the xy plane. If center of C lie [#permalink] New post 24 Aug 2009, 17:48
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Circle C and line k lie in the xy plane. If center of C lie on 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 the line is -1/10.

OPEN DISCUSSION OF THIS TOPIC IS HERE: circle-c-and-line-k-lie-in-the-xy-plane-if-circle-c-is-101471.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 08 Aug 2012, 05:16, edited 1 time in total.
OA added.
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Re: Circle C and line k lie in the xy plane [#permalink] New post 24 Aug 2009, 18:28
C

You can get the answer by drawing a graph.
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Re: Circle C and line k lie in the xy plane [#permalink] New post 24 Aug 2009, 20:29
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I am getting answer E.

Obviously each statement alone is not sufficient.

Combined analysis:
y=-1/10x+a

Finding x intercept:
0=-1/10x+a
x=10a

So two points on the line are (10a,0) and (0,a)

from statement 1, it follows that a>1. Assume that a=1, then the points on a line are (10,0) and (0,1). This line with a=1 intercepts the circle at 2 points. Since it intercepts the circle at 2 points, there must be other lines with a>1 that still intercept the circle. Howerver, as value of a increases, a line shifts to the right and up...eventually it will be tangent to the circle and then outside the circle...hard to explain without graph...
Since we do not know the value of a, three options possible for a>1:
1) the line intercepts the circle at 2 points
2) the line is tangent to the circle
3) the line is outside the circle.
Since we do not know the exact value for a, we can't determine if the line intercepts the circle. I would say both statement are not sufficient - answer E

Wondering if the answer is E?
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Re: Circle C and line k lie in the xy plane [#permalink] New post 24 Aug 2009, 23:03
I too get E, of course without any calculations, just by imagination.

Line with a certain slope can be drawn anywhere in the plane. So it may or may not intersect the circle with radius 1 centered at origin. Only if we had a specific x-intercept we could have concluded that the line intersects the circle or not. Here the x-intercept can be 1,2,3... which has a higher probability to intersect the circle or even 200 which will not intersect the circle.

So E for me. OA?
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Re: Circle C and line k lie in the xy plane [#permalink] New post 25 Aug 2009, 04:06
What is an official answer?
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Re: Circle C and line k lie in the xy plane [#permalink] New post 25 Aug 2009, 08:44
Economist wrote:
I too get E, of course without any calculations, just by imagination.

Line with a certain slope can be drawn anywhere in the plane. So it may or may not intersect the circle with radius 1 centered at origin. Only if we had a specific x-intercept we could have concluded that the line intersects the circle or not. Here the x-intercept can be 1,2,3... which has a higher probability to intersect the circle or even 200 which will not intersect the circle.

So E for me. OA?


But you first have to make sure that the line with the y intercep set as a limit (in this case a=1) is not a tangent line or does not lie entirely outside the circle. If you are given a=X_1 and the line with this y intercept is tangent to the circle, then:
if a=x_1<0, any line with a parameter a<x_1 will be outside the circle
if a=X_1>0, any line with a parameter a>x_1 will be outside the circle.
You do not have necessarily to know the exact value of the intercepts but in this case the answer is E because a line with the y intercept = 1 crosses the circle in 2 points. Hence for any value of y intercept >1, a line may cross the circle at 2 point, be tangent or lie outside the circle.
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Re: Circle C and line k lie in the xy plane [#permalink] New post 25 Aug 2009, 09:13
IMO E. As x intercept is greater than 1 there is possibility for the line to intersect and not to intersect.
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Re: Circle C and line k lie in the xy plane [#permalink] New post 25 Aug 2009, 09:24
sbasha wrote:
Circle C and line k lie in the xy plane. If center of C lie on origin and has radius 1, does line k intersect circle C ?

a. the x intercept of line k is greater than 1.
b. the slope of the line is -1/10.


each alone is not suff

wcs assume x intercept is 1 ( 1,0) is the point

y = -1/10x+b , put x = 1,y = 0 thus b = 1/10 thus k intersect the circle. and by using values for x intercept >1 we can get lines that doesnt intersect...hence E
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Re: Circle C and line k lie in the xy plane [#permalink] New post 27 Aug 2009, 19:07
Thanks for the replies.

OA is E
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Re: Circle C and line k lie in the xy plane [#permalink] New post 27 Aug 2009, 19:22
is there any value of the slope (aside from 0) that would make this sufficient?
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Re: Circle C and line k lie in the xy plane [#permalink] New post 27 Aug 2009, 19:32
Expert's post
sfeiner wrote:
is there any value of the slope (aside from 0) that would make this sufficient?

infinity. (but we still need first statement.) Although I'm doubt about use "infinity" slop term for vertical lines.
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Re: Circle C and line k lie in the xy plane [#permalink] New post 03 Nov 2009, 02:16
I got the correct answer but am not too sure..
the slope is given at -1/10..what does this say?That the slope is -ive so passes through quad 2 and 4...

but what about the inclination?if its almost parallel to the x-axis it may intersect the circle..not so if the line is almost parallel to y axis..

Basically does the figure of 1/10 tell us anything?What if it was given -5?Would the answer be diff.?

Would like to get the answer without calculations:)Please help.
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Re: Circle C and line k lie in the xy plane [#permalink] New post 07 Nov 2009, 17:38
LenaA wrote:
I am getting answer E.

Obviously each statement alone is not sufficient.

Combined analysis:
y=-1/10x+a

Finding x intercept:
0=-1/10x+a
x=10a

So two points on the line are (10a,0) and (0,a)

from statement 1, it follows that a>1. Assume that a=1, then the points on a line are (10,0) and (0,1). This line with a=1 intercepts the circle at 2 points. Since it intercepts the circle at 2 points, there must be other lines with a>1 that still intercept the circle. Howerver, as value of a increases, a line shifts to the right and up...eventually it will be tangent to the circle and then outside the circle...hard to explain without graph...
Since we do not know the value of a, three options possible for a>1:
1) the line intercepts the circle at 2 points
2) the line is tangent to the circle
3) the line is outside the circle.
Since we do not know the exact value for a, we can't determine if the line intercepts the circle. I would say both statement are not sufficient - answer E

Wondering if the answer is E?


Thanks for the detailed explaination. :-D
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Re: Circle C and line k lie in the xy plane [#permalink] New post 29 Mar 2011, 06:54
I made a mistake coz I read the x intercept to be 2, don't ask me how :)
--
Anyhow, for those who cannot visualize, here is an excellent free tool
http://graph-plotter.cours-de-math.eu/

Once we increase the x intercept, the graph will star moving up, therefore the correct asnwer is E.
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Re: Circle C and line k lie in the xy plane [#permalink] New post 19 May 2011, 01:25
a+b
unless and untill y intercept is known line may/may not touch the circle.

y/x = -1/10 touches the circle.

y/x = -20/200 does not touch.

Hence E
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Re: Circle C and line k lie in the xy plane [#permalink] New post 19 May 2011, 14:11
The line x=1 is tangent to the circle at x axis. Any verticale line x>1 would not touch the circle. Any line that is not verticle with x interecept of 1 will intersect the circle at two different places. Since we have a line with -1/10 slope and x interecept that can be very close 1, it might or might not touch based on it's x interecept. Not suff!
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Re: Circle C and line k lie in the xy plane. If center of C lie [#permalink] New post 22 Dec 2011, 10:23
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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.


Line K will intersect circle C under the following minimum conditions


The x-intercept lies between -1 and 1.

-1 < -b/m < 1

--or--

The y-intercept lies between 1 and -1.

1 > b > -1

Statement 1 allows us to eliminate the first condition. Through the x-intercept formula.
Statement 2 does not provide any information about the intercepts.

Taken together, the x-intercept formula allows the deduction

b/.1 > 1

b > .1

that the y-intercept is greater than .1

Is b < 1?
Re: Circle C and line k lie in the xy plane. If center of C lie   [#permalink] 22 Dec 2011, 10:23
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