Author 
Message 
TAGS:

Hide Tags

Intern
Joined: 24 Jan 2012
Posts: 9

A circle with center O and radius 5 is shown in the xyplane
[#permalink]
Show Tags
Updated on: 01 Nov 2014, 06:36
Question Stats:
73% (02:04) correct 27% (02:16) wrong based on 886 sessions
HideShow timer Statistics
A circle with center O and radius 5 is shown in the xyplane. Lines that intersect the circle in 2 points include which of the following ? I. y = x +1 II. y = 2x + 1 III. y = (1/2)x  6 A. I only B. II only C. I and II only D. I and III only E. I, II and III
Official Answer and Stats are available only to registered users. Register/ Login.
Attachments
Circle.jpg [ 8.5 KiB  Viewed 44795 times ]
Originally posted by violetsplash on 26 Sep 2013, 06:25.
Last edited by Bunuel on 01 Nov 2014, 06:36, edited 3 times in total.
RENAMED THE TOPIC.




Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 10105
Location: Pune, India

Re: A circle with center O and radius 5 is shown in the xyplane
[#permalink]
Show Tags
09 Jun 2014, 20:58
violetsplash wrote: A circle with center O and radius 5 is shown in the xyplane. Lines that intersect the circle in 2 points include which of the following ?
I. y = x +1 II. y = 2x + 1 III. y = (1/2)x  6
a) I only b) II Only c) I and II Only d) I and III only e) I, II and III
Please see the attached picture Coordinate geometry is best done by making diagrams of your own. On the x axis, draw a circle with radius 5 and center at (0, 0). It will cut the x axis at 5 and 5 and y axis at 5 and 5. Now try to plot the 3 given lines. I. y = x +1 Put x = 0, you get y = 1. So this line cuts the y axis at 1. Put y = 0, you get x = 1. So this line cuts the x axis at 1. This line, when extended on both sides will cut the circle at two distinct points. II. y = 2x + 1 Similarly, plot this line and you will see that it will cut the circle at two points too. III. y = (1/2)x  6 This line cuts the y axis at 6 and x axis at 12. It is outside the circle and hence doesn't cut the circle. Hence correct answer is (C)
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >




Director
Status: Been a long time guys...
Joined: 03 Feb 2011
Posts: 968
Location: United States (NY)
Concentration: Finance, Marketing
GPA: 3.75

Re: A circle with center O and radius 5 is shown in the xyplane
[#permalink]
Show Tags
26 Sep 2013, 08:16
violetsplash wrote: A circle with center O and radius 5 is shown in the xyplane. Lines that intersect the circle in 2 points include which of the following ?
I. y = x +1 II. y = 2x + 1 III. y = (1/2)x  6
a) I only b) II Only c) I and II Only d) I and III only e) I, II and II
Find the intercepts by making once x=0 and then y=0. On finding these intercepts, draw line joining these points. ex for line y=x+1, the points will be (1,0) and (0,1). On joining these two points, it gets clear that the line will intersect the circle at two points. Similarly for II and III. You will see that III is far away from the circle. Hence I and II cut the circle. +1C
_________________




Math Expert
Joined: 02 Sep 2009
Posts: 61281

Re: A circle with center O and radius 5 is shown in the xyplane
[#permalink]
Show Tags
26 Sep 2013, 07:08
violetsplash wrote: A circle with center O and radius 5 is shown in the xyplane. Lines that intersect the circle in 2 points include which of the following ?
I. y = x +1 II. y = 2x + 1 III. y = (1/2)x  6
a) I only b) II Only c) I and II Only d) I and III only e) I, II and III
Please see the attached picture Check the diagram below: Attachment:
graph.png [ 9.35 KiB  Viewed 44672 times ]
Answer: C.
_________________



Verbal Forum Moderator
Joined: 10 Oct 2012
Posts: 570

Re: A circle with center O and radius 5 is shown in the xyplane
[#permalink]
Show Tags
26 Sep 2013, 08:41
violetsplash wrote: A circle with center O and radius 5 is shown in the xyplane. Lines that intersect the circle in 2 points include which of the following ?
I. y = x +1 II. y = 2x + 1 III. y = (1/2)x  6
a) I only b) II Only c) I and II Only d) I and III only e) I, II and III
Please see the attached picture If a line is intersecting a circle at 2 places, the distance between that line and the center of the circle< The radius Also, we know that the distance between a point (p,q) and the line ax+by+c=0 is \(\frac{ap+qb+c}{\sqrt{a^2+b^2}\) It's very easy to realise that (p,q) = (0,0) and the distance between the lines in I. and II. are less than 5. For III. the distance =\(\frac{6}{\sqrt{\frac{5}{4}}\) = \(\frac{12}{\sqrt{5}}\)>5 Thus, line in option III. doesn't intersect the circle at all. C
_________________



Intern
Joined: 22 Nov 2010
Posts: 4

Re: A circle with center O and radius 5 is shown in the xyplane
[#permalink]
Show Tags
09 Jun 2014, 19:58
Bunuel wrote: violetsplash wrote: A circle with center O and radius 5 is shown in the xyplane. Lines that intersect the circle in 2 points include which of the following ?
I. y = x +1 II. y = 2x + 1 III. y = (1/2)x  6
a) I only b) II Only c) I and II Only d) I and III only e) I, II and III
Please see the attached picture Check the diagram below: Attachment: graph.png Answer: C. Hi Bunuel, sorry I did not get this around in my head... Could you explain in a bit more detail please? I am bad with coordinate geometry, these things just dont seem to get into my head



Director
Joined: 26 Aug 2014
Posts: 824
Concentration: Marketing
Schools: HBS '17 (S), Kellogg '17 (M), Booth '17 (A), CBS '17 (D), Insead Sept '16 (D), HEC Sept'16 (A), CnsrtmAnderson '17 (A$), CnsrtmStern '17 (D), CnsrtmKenanFlagler '17 (A$), CnsrtmMarshall '17 (A), CnsrtmYale '17 (D), CnsrtmRoss '17 (A$)
GPA: 3.4

Re: A circle with center O and radius 5 is shown in the xyplane
[#permalink]
Show Tags
01 Nov 2014, 06:28
i did this a much faster way (hopefully it works in all aspects).
I mad x 0 for every equation to find the y intercept.
I. Y = 1 II y = 1 III Y  6
The circle only spans from 5 to 5 so only I and 2 will intersect in 2 points.



Intern
Joined: 21 Feb 2015
Posts: 24

A circle with center O and radius 5 is shown in the xyplane
[#permalink]
Show Tags
02 Apr 2015, 17:48
I understand the explanation, but not the question being ask by GMAT. " Lines that intersect the circle in 2 points include which of the following ?" is this question asking us identify the lines that intersect the circumference of the circle twice? Or the question is asking us to identify the lines that has two points within the circumference of the circle?



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 10105
Location: Pune, India

Re: A circle with center O and radius 5 is shown in the xyplane
[#permalink]
Show Tags
02 Apr 2015, 18:29
mawus wrote: I understand the explanation, but not the question being ask by GMAT. " Lines that intersect the circle in 2 points include which of the following ?" is this question asking us identify the lines that intersect the circumference of the circle twice? Or the question is asking us to identify the lines that has two points within the circumference of the circle? A line that intersects a circle in 2 points is one which has 2 points lying on the circumference of the circle (so it intersects the circumference twice i.e. to say it is not a tangent).
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >



Intern
Joined: 21 Feb 2015
Posts: 24

A circle with center O and radius 5 is shown in the xyplane
[#permalink]
Show Tags
Updated on: 03 Apr 2015, 05:39
Hi Karishma, What if the first line,y = x +1, has a restriction that x and y< +5. Is it still going to qualify as a one of the sufficient answer choices?
Originally posted by Rookie124 on 03 Apr 2015, 05:21.
Last edited by Rookie124 on 03 Apr 2015, 05:39, edited 1 time in total.



Intern
Joined: 14 Oct 2013
Posts: 42

Re: A circle with center O and radius 5 is shown in the xyplane
[#permalink]
Show Tags
16 May 2015, 06:26
Why do we assume the center is 0,0? I understand that the radius is 5 so it makes logical sense that the center is 0,0 but couldn't the center easily be for example (0,2) and then the top point of the circle would be (0,3) and bottom be (0,7). Radius is still 5 but the circle just sits in a different place in the coordinate plane. Posted from GMAT ToolKit



EMPOWERgmat Instructor
Status: GMAT Assassin/CoFounder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 16107
Location: United States (CA)

Re: A circle with center O and radius 5 is shown in the xyplane
[#permalink]
Show Tags
16 May 2015, 15:17
Hi healthjunkie, The original prompt includes a picture that places O at the Origin. IF that drawing was NOT included with the question, and it wasn't clear that the circle was centered at the Origin, then your concerns would be valid. Having the picture to work with, are you comfortable answering the question? GMAT assassins aren't born, they're made, Rich
_________________
Contact Rich at: Rich.C@empowergmat.comThe Course Used By GMAT Club Moderators To Earn 750+ souvik101990 Score: 760 Q50 V42 ★★★★★ ENGRTOMBA2018 Score: 750 Q49 V44 ★★★★★



Intern
Joined: 14 Oct 2013
Posts: 42

Re: A circle with center O and radius 5 is shown in the xyplane
[#permalink]
Show Tags
17 May 2015, 15:11
Ah, I didnt see the picture as I was doing the question on my phone and somehow I don't think the picture loaded. Makes sense now thanks!



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 10105
Location: Pune, India

Re: A circle with center O and radius 5 is shown in the xyplane
[#permalink]
Show Tags
18 May 2015, 21:48
mawus wrote: Hi Karishma, What if the first line,y = x +1, has a restriction that x and y< +5. Is it still going to qualify as a one of the sufficient answer choices? I am not sure I understand your question. y = x + 1 is the equation of a line (which is infinite on both ends) and it intersects the given circle at two points (4, 3) and (3, 4). Absolute values of both x and y coordinates are less than 5 (if that's what you meant). The line intersects the circle in two points and hence will be a part of the answer.
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >



Intern
Joined: 09 Dec 2013
Posts: 22
Concentration: Entrepreneurship, Finance
GMAT 1: 570 Q45 V24 GMAT 2: 660 Q47 V34
GPA: 3.36
WE: Engineering (Manufacturing)

Re: A circle with center O and radius 5 is shown in the xyplane
[#permalink]
Show Tags
10 Jul 2016, 06:54
Hi,
I understand the approaches provided by experts.
However, in the real test with scrap paper, how can we draw a perfect circle?
The line in choice iii could have intersected the circle just a tiny bit.
Thanks,



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 10105
Location: Pune, India

Re: A circle with center O and radius 5 is shown in the xyplane
[#permalink]
Show Tags
11 Jul 2016, 21:22
Danuthan wrote: Hi,
I understand the approaches provided by experts.
However, in the real test with scrap paper, how can we draw a perfect circle?
The line in choice iii could have intersected the circle just a tiny bit.
Thanks, In case of a doubt, just find the shortest distance of the line from the centre (0, 0). The circle with radius 5 will have every point at a distance of 5 from (0, 0). If the shortest distance of the line from (0, 0) is more than 5, it will not cut the circle at all. The line intersects the x axis at 12 and y axis at 6. So it will form a right triangle with the axis such that hypotenuse is \(\sqrt{12^2 + 6^2} = \sqrt{180} = 6*\sqrt{5}\) Area of the triangle = (1/2)*Leg1*Leg2 = (1/2)*Altitude * Hypotenuse \(6*12 = Altitude * 6*\sqrt{5}\) \(Altitude = 2.4*\sqrt{5} = 2.4*2.2 = 5.3 (approx)\) So shortest distance of the line from (0, 0) is 5.3 which is greater than 5. The line doesn't intersect the circle.
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >



Intern
Joined: 23 Sep 2016
Posts: 5

A circle with center O and radius 5 is shown in the xyplane
[#permalink]
Show Tags
09 Oct 2016, 13:28
vikramjainbus137 wrote: Bunuel wrote: violetsplash wrote: A circle with center O and radius 5 is shown in the xyplane. Lines that intersect the circle in 2 points include which of the following ?
I. y = x +1 II. y = 2x + 1 III. y = (1/2)x  6
a) I only b) II Only c) I and II Only d) I and III only e) I, II and III
Please see the attached picture Check the diagram below: Attachment: graph.png Answer: C. Hi Bunuel, sorry I did not get this around in my head... Could you explain in a bit more detail please? I am bad with coordinate geometry, these things just dont seem to get into my head The way Bunuel plotted the points, drew the lines, and got the answer, without having to manually calculate anything was by looking at the yintercept and the slope: I. y= x + 1 ... AKA y = \(\frac{1}{1}\)x+ 1 Y int = +1 Slope = \(\frac{1}{1}\)(slope is the number before "x"... in this case it's 1) 1) draw a point at your yint of (0,1) 2) starting from your Yintercept, which in this case is (0,1), you have to RISE 1, RUN 1 ... that's your second point .... then RISE 1, RUN 1 again and that's your third point on the line, and so on... then connect the points and you have your line with slope of 1/1... this line intercepts the given circle at 2 points II. y= 2x + 1 Y int = +1 Slope = \(\frac{+2}{1}\) 1) Draw a point at (0,1) 2) Starting from (0,1) RISE 2, RUN 1 and so on, like you did for number I above. III. y = \(\frac{1}{2}\)x  6 Y int = 6 Slope = \(\frac{+1}{2}\) 1) Draw a point at (0,6) 2) Starting from (0,6), RISE 1, RUN 2 and so on Number III is the only one that doesn't intersect the circle in 2 points... so answer is C.



Manager
Joined: 13 Dec 2013
Posts: 141
Location: United States (NY)
Concentration: General Management, International Business
GMAT 1: 710 Q46 V41 GMAT 2: 720 Q48 V40
GPA: 4
WE: Consulting (Consulting)

Re: A circle with center O and radius 5 is shown in the xyplane
[#permalink]
Show Tags
01 Apr 2017, 12:25
VeritasPrepKarishma wrote: violetsplash wrote: A circle with center O and radius 5 is shown in the xyplane. Lines that intersect the circle in 2 points include which of the following ?
I. y = x +1 II. y = 2x + 1 III. y = (1/2)x  6
a) I only b) II Only c) I and II Only d) I and III only e) I, II and III
Please see the attached picture Coordinate geometry is best done by making diagrams of your own. On the x axis, draw a circle with radius 5 and center at (0, 0). It will cut the x axis at 5 and 5 and y axis at 5 and 5. Now try to plot the 3 given lines. I. y = x +1 Put x = 0, you get y = 1. So this line cuts the y axis at 1. Put y = 0, you get x = 1. So this line cuts the x axis at 1. This line, when extended on both sides will cut the circle at two distinct points. II. y = 2x + 1 Similarly, plot this line and you will see that it will cut the circle at two points too. III. y = (1/2)x  6 This line cuts the y axis at 6 and x axis at 12. It is outside the circle and hence doesn't cut the circle. Hence correct answer is (C) For I and II, calculating both points isn't necessary is it? If one point is in the circle then the line will have to intersect the circle at two points. However, if a point was touching the circumference, then determination of the second point would be necessary.



Manager
Joined: 07 Feb 2017
Posts: 172

Re: A circle with center O and radius 5 is shown in the xyplane
[#permalink]
Show Tags
13 Jun 2018, 13:58
I and II Only Check if point on line exists inside the circle Check x = 0 I. y = 1 yes II. y = 1 yes
III. y = 6 Check the slope It's 1/2 so line is flatter than slope 1; line has point below circle and slanted downward from circle



Director
Joined: 04 Aug 2010
Posts: 532
Schools: Dartmouth College

Re: A circle with center O and radius 5 is shown in the xyplane
[#permalink]
Show Tags
13 Jun 2018, 15:23
violetsplash wrote: A circle with center O and radius 5 is shown in the xyplane. Lines that intersect the circle in 2 points include which of the following ?
I. y = x +1 II. y = 2x + 1 III. y = (1/2)x  6
A. I only B. II only C. I and II only D. I and III only E. I, II and III Any line that passes through the interior of the circle will intersect the circle at two points. The xintercepts of the circle are 5 and 5. The yintercepts of the circle are 5 and 5. I. y = x + 1 II. y = 2x + 1 Each of these lines has a yintercept between 5 and 5. Thus, each line must pass through the interior of the circle. Eliminate any answer choice that does not include both I and II (A, B and D). III. y = (1/2)x  6 Here, the yintercept = 6. To determine the xintercept, substitute y=0 and solve for x: 0 = (1/2)x  6 x=12. Since the xintercept = 12 and the yintercept = 6, this line does not pass through the interior of the circle. Eliminate any remaining answer choice that includes III (E).
_________________
GMAT and GRE Tutor New York, NY
Available for tutoring in NYC and longdistance. For more information, please email me at GMATGuruNY@gmail.com.




Re: A circle with center O and radius 5 is shown in the xyplane
[#permalink]
13 Jun 2018, 15:23



Go to page
1 2
Next
[ 27 posts ]



