GMAT Question of the Day - Daily to your Mailbox; hard ones only

It is currently 07 Dec 2019, 22:47

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

A circle with center O and radius 5 is shown in the xy-plane

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Find Similar Topics 
Intern
Intern
avatar
Joined: 24 Jan 2012
Posts: 9
A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

Show Tags

New post Updated on: 01 Nov 2014, 07:36
11
41
00:00
A
B
C
D
E

Difficulty:

  35% (medium)

Question Stats:

73% (02:02) correct 27% (02:16) wrong based on 829 sessions

HideShow timer Statistics

A circle with center O and radius 5 is shown in the xy-plane. 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

Attachments

Circle.jpg
Circle.jpg [ 8.5 KiB | Viewed 43129 times ]


Originally posted by violetsplash on 26 Sep 2013, 07:25.
Last edited by Bunuel on 01 Nov 2014, 07:36, edited 3 times in total.
RENAMED THE TOPIC.
Most Helpful Expert Reply
Veritas Prep GMAT Instructor
User avatar
V
Joined: 16 Oct 2010
Posts: 9850
Location: Pune, India
Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

Show Tags

New post 09 Jun 2014, 21:58
19
1
10
violetsplash wrote:
A circle with center O and radius 5 is shown in the xy-plane. 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


Co-ordinate 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 >
Most Helpful Community Reply
Director
Director
User avatar
Status: Been a long time guys...
Joined: 03 Feb 2011
Posts: 987
Location: United States (NY)
Concentration: Finance, Marketing
GPA: 3.75
GMAT ToolKit User
Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

Show Tags

New post 26 Sep 2013, 09:16
13
1
5
violetsplash wrote:
A circle with center O and radius 5 is shown in the xy-plane. 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
_________________
General Discussion
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 59588
Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

Show Tags

New post 26 Sep 2013, 08:08
14
11
violetsplash wrote:
A circle with center O and radius 5 is shown in the xy-plane. 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
graph.png [ 9.35 KiB | Viewed 43021 times ]
Answer: C.
_________________
Verbal Forum Moderator
User avatar
B
Joined: 10 Oct 2012
Posts: 583
Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

Show Tags

New post 26 Sep 2013, 09:41
6
5
violetsplash wrote:
A circle with center O and radius 5 is shown in the xy-plane. 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
Intern
avatar
Joined: 22 Nov 2010
Posts: 4
Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

Show Tags

New post 09 Jun 2014, 20:58
Bunuel wrote:
violetsplash wrote:
A circle with center O and radius 5 is shown in the xy-plane. 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
Director
avatar
Joined: 26 Aug 2014
Posts: 824
Concentration: Marketing
GPA: 3.4
Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

Show Tags

New post 01 Nov 2014, 07:28
7
1
1
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
Intern
avatar
Joined: 21 Feb 2015
Posts: 24
A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

Show Tags

New post 02 Apr 2015, 18:48
1
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
User avatar
V
Joined: 16 Oct 2010
Posts: 9850
Location: Pune, India
Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

Show Tags

New post 02 Apr 2015, 19:29
1
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
Intern
avatar
Joined: 21 Feb 2015
Posts: 24
A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

Show Tags

New post Updated on: 03 Apr 2015, 06:39
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, 06:21.
Last edited by Rookie124 on 03 Apr 2015, 06:39, edited 1 time in total.
Intern
Intern
avatar
Joined: 14 Oct 2013
Posts: 43
GMAT ToolKit User
Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

Show Tags

New post 16 May 2015, 07:26
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.

Image Posted from GMAT ToolKit
EMPOWERgmat Instructor
User avatar
V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 15669
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

Show Tags

New post 16 May 2015, 16:17
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.com
Image


The Course Used By GMAT Club Moderators To Earn 750+

souvik101990 Score: 760 Q50 V42 ★★★★★
ENGRTOMBA2018 Score: 750 Q49 V44 ★★★★★
Intern
Intern
avatar
Joined: 14 Oct 2013
Posts: 43
GMAT ToolKit User
Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

Show Tags

New post 17 May 2015, 16:11
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
User avatar
V
Joined: 16 Oct 2010
Posts: 9850
Location: Pune, India
Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

Show Tags

New post 18 May 2015, 22:48
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 co-ordinates 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
Intern
avatar
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)
GMAT ToolKit User
Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

Show Tags

New post 10 Jul 2016, 07:54
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
User avatar
V
Joined: 16 Oct 2010
Posts: 9850
Location: Pune, India
Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

Show Tags

New post 11 Jul 2016, 22:22
3
1
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
Intern
avatar
Joined: 23 Sep 2016
Posts: 5
A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

Show Tags

New post 09 Oct 2016, 14:28
vikramjainbus137 wrote:
Bunuel wrote:
violetsplash wrote:
A circle with center O and radius 5 is shown in the xy-plane. 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 y-intercept 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 y-int of (0,1)
2) starting from your Y-intercept, 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
Manager
avatar
B
Joined: 13 Dec 2013
Posts: 142
Location: United States (NY)
Concentration: General Management, International Business
Schools: Cambridge"19 (A)
GMAT 1: 710 Q46 V41
GMAT 2: 720 Q48 V40
GPA: 4
WE: Consulting (Consulting)
Reviews Badge
Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

Show Tags

New post 01 Apr 2017, 13:25
VeritasPrepKarishma wrote:
violetsplash wrote:
A circle with center O and radius 5 is shown in the xy-plane. 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


Co-ordinate 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
Manager
avatar
S
Joined: 07 Feb 2017
Posts: 175
Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

Show Tags

New post 13 Jun 2018, 14:58
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
Senior Manager
Senior Manager
avatar
G
Joined: 04 Aug 2010
Posts: 496
Schools: Dartmouth College
Re: A circle with center O and radius 5 is shown in the xy-plane  [#permalink]

Show Tags

New post 13 Jun 2018, 16:23
2
violetsplash wrote:
A circle with center O and radius 5 is shown in the xy-plane. 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 x-intercepts of the circle are -5 and 5.
The y-intercepts of the circle are -5 and 5.

I. y = -x + 1
II. y = 2x + 1
Each of these lines has a y-intercept 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 y-intercept = -6.
To determine the x-intercept, substitute y=0 and solve for x:
0 = (1/2)x - 6
x=12.
Since the x-intercept = 12 and the y-intercept = -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 long-distance.
For more information, please email me at GMATGuruNY@gmail.com.
GMAT Club Bot
Re: A circle with center O and radius 5 is shown in the xy-plane   [#permalink] 13 Jun 2018, 16:23

Go to page    1   2    Next  [ 27 posts ] 

Display posts from previous: Sort by

A circle with center O and radius 5 is shown in the xy-plane

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne