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Manager  Joined: 23 Sep 2009
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Circle C and line K lie in the XY plane. If circle C is centered at th  [#permalink]

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30 00:00

Difficulty:   65% (hard)

Question Stats: 57% (01:54) correct 43% (01:53) wrong based on 430 sessions

### HideShow timer Statistics 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

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Originally posted by vigneshpandi on 21 Sep 2010, 21:28.
Last edited by Bunuel on 02 Apr 2019, 03:36, edited 2 times in total.
Edited the question
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Re: Circle C and line K lie in the XY plane. If circle C is centered at th  [#permalink]

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10
5
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. For more on this issue check Coordinate Geometry Chapter of Math Book: http://gmatclub.com/forum/math-coordina ... 87652.html

Hope it helps.

Attachment: graph.png [ 15.11 KiB | Viewed 19517 times ]

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Manager  Joined: 23 Sep 2009
Posts: 105
Re: Circle C and line K lie in the XY plane. If circle C is centered at th  [#permalink]

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Thank you for the explanation....And the Math book is really helpfull.
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Re: Circle C and line K lie in the XY plane. If circle C is centered at th  [#permalink]

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anilnandyala wrote:
circle c & line k lie in xy-plane. if circle c is centered at the origin & has a radius 1 . does line k intersect circle c?

1 the x intercept of line k>1
2 the slope of line k= -(1/10)

Look at the diagrams.

Statement 1: If x intercept > 1, the line can be any of the following (and can be drawn in many more ways)
Attachment: Ques.jpg [ 16.54 KiB | Viewed 18243 times ]

Statement 2: If slope = -1/10, the line can be drawn in any of the following ways. (and many more)
Attachment: Ques1.jpg [ 14.73 KiB | Viewed 18238 times ]

Using both together: Look at the diagrams above. Both have x intercepts greater than 1 and slope = -1/10. In one case, it will intersect the circle, in the other case, it will not. SO both statements together are not sufficient. Answer (E).
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GMAT 1: 780 Q51 V48 GRE 1: Q800 V740 Re: Circle C and line K lie in the XY plane. If circle C is centered at th  [#permalink]

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Using statement 1: If the X-intercept of line K is greater than 1, it may or not intersect circle C. For example, if the line is x=3 then it does not intersect the circle C, but if the line is y = -x/4 + 1/2 then it has x-intercept = 2 (>1) and does intersect the circle C. Therefore statement (1) is insufficient.

Using statement 2: If the slope of the line is -1/10, then it may or may not intersect the circle C. For example, y= -x/10 + 2 does not intersect the circle, but y = -x/10 + 0.5 does intersect the circle, though both have a slope of -1/10. Therefore statement 2 is insufficient to answer the question.

Combining statements 1 and 2, the statements together are still insufficient to answer the question. For example, y = -x/10 + 0.5 has an x-intercept greater than 1 and a slope of -1/10, but it does intersect the circle C. However, y = -x/10 + 2 also has an x-intercept greater than 1 and a slope of -1/10, but it does not intersect the circle C.

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

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2
Circle C and line K lie in XY plane. If circle C is centered at the origin and has radius 1, does line K intersect circle C?
1) X intercept of line K is greater than 1
2) The slope of like K is -1/10

Draw some diagrams to figure it out. Even using both statements, you get two cases - one in which K intersects C and another in which it doesn't.

Attachment: Ques4.jpg [ 9.68 KiB | Viewed 18229 times ]

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

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a. y=mx+c so for x= -c/m where m can be 1 or -1. c can be 0.1 or 100.

hence a is POE thus D is POE too.

similarly for b too 'c' can have any such values as a.

thus POE.

a+b

y= -0.1x +c meaning x= c/0.1 > 1 thus c > 0.1
hence y intercept can be 0.1 with a slope of -0.1
or 100 with the same slope. not sufficient.

E it is.
Manager  Joined: 12 Oct 2011
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Re: Circle C and line K lie in the XY plane. If circle C is centered at th  [#permalink]

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E it is. The diagramming method as shown by Karishma worked best for me.

Statement 1 is insufficient as it gives only x intercept. A line with an x-intercept > 1 may or may not pass through the circle given. INSUFFICIENT.

Statement 2 gives a negative slope. The line is decreasing as it goes from left to right but it does not give an indication whether the line intersects the circle or not. INSUFFICIENT.

Even taken together, two lines can be drawn both with x-intercept more than one and with given negative slope. One of the lines may intersect the circle and the other may not. INSUFFICIENT.

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GMAT 1: 780 Q51 V48 GRE 1: Q800 V740 Re: Circle C and line K lie in the XY plane. If circle C is centered at th  [#permalink]

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We can see that the circle is centered at origin with radius 1 and has the equation x^2 + y^2 = 1

Using statement (1): If the x-intercept of line k is greater than 1, it may or may not intersect the circle. The line can have any slope and any x-intercept and not all of such lines will intersect with the circle. Insufficient.

Using statement (2): Just knowing the slope without any point that the line passes through is insufficient as with a particular slope infinite lines can be drawn that do and do not intercept the circle. Insufficient.

Combining statement (1) and (2), we get a definite equation of one unique line as we know the x-intercept and the slope. Therefore we can conclude that this line will either intersect the circle or will not. Sufficient.

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

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Responding to a pm:

Quote:
I have solved the question correctly but in a totally different way. kindly let me know if my process is alright.
statement 1: x=-(c/m)>1
hence slope is less than 0 so the line may or may not touch the circle.

Statement 2: slope is negative hence the line is downward moving. thus it can or cannot touch the circle.

Combining the statement no new info found hence E.

Plus can you please give me some insights on the concept of intercepts?

How do you figure that slope is negative from statement 1?

Slope = - y intercept/x intercept
So you get
m = -c/x (correct)
x = -c/m > 1

Here is my problem: why do you say that m must be negative? c could be negative instead.

Though using both statements, we know that x intercept is greater than 1 and the slope is negative.
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Re: Circle C and line K lie in the XY plane. If circle C is centered at th  [#permalink]

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(1) The x-intercept of line K is greater than 1 implies that the x-intercept of line K is towards the right of circle C but this does not imply if line K intersects circle C; NOT sufficient.

(2) The slope of line K is -1/10 implies line K goes down left to right, but again this does not imply if line K intersects circle C; NOT sufficient.

Combining (1) and (2), we do not know the exact intercept of line K, and we cannot find if line K intersects circle C or not; NOT sufficient.

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

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vigneshpandi 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

we do not to draw many things.

if k is perpendicular to x axis , k must be paralel to y axis. k have negative slope this mean k can not parallel to y, so if x intercept is greater than 1, k can cut the circle.

if we move k to the right, k can not cut circle
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Re: Circle C and line K lie in the XY plane. If circle C is centered at th  [#permalink]

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