Draw a Diagram of a Circle in the X-y Plane

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A circle with center O and radius v is shown in the xy-airplane [#permalink]   Updated on: 01 Nov 2014, 06:36

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Question Stats:

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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 + i
III. y = (1/ii)10 - 6

A. I merely
B. 2 only
C. I and 2 merely
D. I and III just
E. I, II and Iii

Attachments

Circumvolve.jpg [ 8.5 KiB | Viewed 63727 times ]

Originally posted by violetsplash on 26 Sep 2013, 06:25.
Last edited past Bunuel on 01 Nov 2014, 06:36, edited iii times in total.

RENAMED THE TOPIC.

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Re: A circumvolve with center O and radius 5 is shown in the xy-plane [#permalink]   09 Jun 2014, 20:58

violetsplash wrote:

A circle with heart O and radius v 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 = (ane/2)10 - 6

a) I merely
b) II Only
c) I and Ii Only
d) I and III only
e) I, Ii and Three

Please see the attached pic

Co-ordinate geometry is best done past making diagrams of your ain. On the x axis, draw a circle with radius 5 and center at (0, 0). It will cut the 10 axis at 5 and -five and y centrality at 5 and -5.

Now endeavour to plot the 3 given lines.

I. y = -x +1
Put x = 0, you become y = 1. So this line cuts the y axis at one.
Put y = 0, you go x = 1. So this line cuts the x centrality at 1.
This line, when extended on both sides volition cutting the circle at two distinct points.

Two. y = 2x + 1
Similarly, plot this line and you lot will see that information technology will cutting the circle at two points too.

Three. y = (1/2)x - half-dozen
This line cuts the y axis at -6 and ten axis at 12. It is outside the circle and hence doesn't cut the circle.

Hence correct respond is (C)
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Re: A circle with center O and radius 5 is shown in the xy-plane [#permalink]   26 Sep 2013, 08:16

violetsplash wrote:

A circumvolve with center O and radius v is shown in the xy-plane. Lines that intersect the circumvolve in two points include which of the post-obit ?

I. y = -10 +1
II. y = 2x + i
III. y = (1/ii)x - half dozen

a) I only
b) II Only
c) I and II Only
d) I and 3 only
eastward) I, II and Ii

Discover the intercepts by making one time 10=0 and then y=0. On finding these intercepts, draw line joining these points.
ex for line y=-ten+1, the points will be (1,0) and (0,1). On joining these two points, it gets clear that the line volition intersect the circle at two points. Similarly for II and Iii.
You lot volition come across that III is far away from the circumvolve.
Hence I and Two cut the circle.
+1C
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Re: A circle with center O and radius 5 is shown in the xy-plane [#permalink]   26 Sep 2013, 07:08

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 = -ten +1
Two. y = 2x + 1
III. y = (1/2)ten - half dozen

a) I simply
b) II Only
c) I and II Merely
d) I and 3 merely
e) I, II and Three

Please see the attached movie

Check the diagram below:

Zipper:

graph.png [ 9.35 KiB | Viewed 62490 times ]

Answer: C.
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Re: A circle with center O and radius 5 is shown in the xy-airplane [#permalink]   26 Sep 2013, 08:41

violetsplash wrote:

A circle with center O and radius v is shown in the xy-aeroplane. Lines that intersect the circle in 2 points include which of the following ?

I. y = -x +1
II. y = 2x + 1
Iii. y = (1/ii)x - 6

a) I only
b) II Only
c) I and Ii Only
d) I and III merely
e) I, II and 3

Please see the attached film

If a line is intersecting a circle at 2 places, the distance between that line and the center of the circle< The radius

As well, nosotros know that the distance betwixt a point (p,q) and the line ax+by+c=0 is \(\frac{|ap+qb+c|}{\sqrt{a^two+b^2}\)

It's very like shooting fish in a barrel to realise that (p,q) = (0,0) and the altitude between the lines in I. and 2. are less than 5. For III. the distance =\(\frac{|half-dozen|}{\sqrt{\frac{5}{4}}\) = \(\frac{12}{\sqrt{5}}\)>5

Thus, line in option III. doesn't intersect the circumvolve at all.

C
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Re: A circle with middle O and radius 5 is shown in the xy-plane [#permalink]   09 Jun 2014, 19:58

Bunuel wrote:

violetsplash wrote:

A circumvolve with center O and radius five is shown in the xy-plane. Lines that intersect the circle in ii points include which of the post-obit ?

I. y = -10 +1
II. y = 2x + ane
III. y = (1/two)x - six

a) I only
b) 2 Just
c) I and Two Simply
d) I and III only
eastward) I, II and III

Please see the attached picture

Check the diagram below:

Zipper:

graph.png

Answer: C.

How-do-you-do Bunuel, sorry I did not become this around in my head... Could you explain in a bit more detail please?

I am bad with coordinate geometry, these things simply dont seem to get into my head

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Re: A circumvolve with center O and radius 5 is shown in the xy-plane [#permalink]   01 November 2014, 06:28

i did this a much faster way (hopefully it works in all aspects).

I mad x 0 for every equation to observe the y intercept.

I. Y = one
II y = 1
III Y - -half dozen

The circle but spans from -5 to v so only I and 2 volition intersect in ii points.

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A circle with center O and radius 5 is shown in the xy-plane [#permalink]   02 April 2015, 17:48

I empathize the caption, simply not the question being enquire by GMAT. " Lines that intersect the circumvolve in 2 points include which of the following ?" is this question asking u.s. 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?

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Re: A circle with center O and radius five is shown in the xy-plane [#permalink]   02 Apr 2015, 18:29

mawus wrote:

I sympathise the explanation, only non the question being enquire by GMAT. " Lines that intersect the circle in two points include which of the post-obit ?" is this question asking us identify the lines that intersect the circumference of the circle twice? Or the question is asking usa to identify the lines that has two points inside 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).
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A circle with centre O and radius v is shown in the xy-plane [#permalink]   Updated on: 03 Apr 2015, 05:39

Hi Karishma,
What if the showtime line,y = -x +i, has a brake that x and y< +-5. Is it still going to qualify as a i of the sufficient answer choices?

Originally posted past Rookie124 on 03 Apr 2015, 05:21.
Last edited by Rookie124 on 03 Apr 2015, 05:39, edited one time in total.

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Re: A circle with centre O and radius five is shown in the xy-airplane [#permalink]   sixteen May 2015, 06:26

Why practice nosotros presume the centre is 0,0? I empathize that the radius is five and then it makes logical sense that the middle is 0,0 but couldn't the center easily be for case (0,-ii) and so the superlative betoken of the circle would exist (0,three) and bottom exist (0,-7). Radius is notwithstanding 5 simply the circle but sits in a different place in the coordinate plane.

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Re: A circle with middle O and radius five is shown in the xy-airplane [#permalink]   16 May 2015, 15:17

Hello 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 articulate that the circumvolve was centered at the Origin, then your concerns would be valid. Having the picture to piece of work with, are you comfortable answering the question?

GMAT assassins aren't born, they're fabricated,
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Re: A circle with heart O and radius five is shown in the xy-aeroplane [#permalink]   eighteen May 2015, 21:48

mawus wrote:

Howdy Karishma,
What if the first line,y = -x +i, has a restriction that ten and y< +-5. Is it nevertheless going to qualify as a 1 of the sufficient reply 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 information technology intersects the given circle at two points (4, -three) and (-three, iv). Absolute values of both ten and y co-ordinates are less than |v| (if that's what yous meant).
The line intersects the circle in two points and hence will be a part of the answer.
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Re: A circle with center O and radius v is shown in the xy-plane [#permalink]   10 Jul 2016, 06:54

Hi,

I understand the approaches provided by experts.

However, in the real test with fleck paper, how can we draw a perfect circle?

The line in choice iii could take intersected the circumvolve simply a tiny fleck.

Thanks,

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Re: A circumvolve with heart O and radius 5 is shown in the xy-aeroplane [#permalink]   11 Jul 2016, 21:22

Danuthan wrote:

Hi,

I understand the approaches provided by experts.

However, in the real exam with flake newspaper, how can nosotros draw a perfect circle?

The line in option three could have intersected the circle only a tiny bit.

Thanks,

In case of a incertitude, simply detect the shortest distance of the line from the center (0, 0). The circle with radius 5 will accept every point at a altitude of 5 from (0, 0). If the shortest distance of the line from (0, 0) is more than v, it will not cut the circle at all.

The line intersects the x axis at 12 and y centrality at -6. So it will form a right triangle with the centrality such that hypotenuse is
\(\sqrt{12^2 + half-dozen^2} = \sqrt{180} = 6*\sqrt{5}\)

Surface area of the triangle = (1/2)*Leg1*Leg2 = (1/2)*Altitude * Hypotenuse

\(six*12 = Distance * 6*\sqrt{5}\)

\(Altitude = ii.iv*\sqrt{5} = two.iv*2.ii = 5.3 (approx)\)

And then shortest distance of the line from (0, 0) is 5.3 which is greater than v. The line doesn't intersect the circle.
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Re: A circle with center O and radius 5 is shown in the xy-plane [#permalink]   thirteen Jun 2018, 15:23

violetsplash wrote:

A circle with center O and radius 5 is shown in the xy-airplane. Lines that intersect the circle in 2 points include which of the following ?

I. y = -x +i
II. y = 2x + 1
III. y = (1/2)10 - 6

A. I only
B. 2 only
C. I and Ii just
D. I and Iii only
E. I, Two and III

Any line that passes through the interior of the circumvolve will intersect the circle at ii points.
The x-intercepts of the circumvolve are -5 and 5.
The y-intercepts of the circle are -v and v.

I. y = -x + 1
Ii. y = 2x + i
Each of these lines has a y-intercept betwixt -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 = -vi.
To make up one's mind the x-intercept, substitute y=0 and solve for x:
0 = (1/2)10 - half-dozen
x=12.
Since the x-intercept = 12 and the y-intercept = -half dozen, this line does not pass through the interior of the circle.
Eliminate whatsoever remaining answer choice that includes Iii (East).

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Re: A circle with center O and radius five is shown in the xy-aeroplane [#permalink]   20 Jul 2018, 08:29

GMATGuruNY wrote:

violetsplash wrote:

A circle with middle 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 = (i/two)x - 6

A. I only
B. II merely
C. I and II simply
D. I and III but
E. I, Ii and III

Any line that passes through the interior of the circle volition intersect the circle at two points.
The x-intercepts of the circle are -5 and 5.
The y-intercepts of the circle are -five and 5.

I. y = -x + 1
II. y = 2x + 1
Each of these lines has a y-intercept between -v and five.
Thus, each line must laissez passer through the interior of the circle.
Eliminate any answer choice that does not include both I and II (A, B and D).

3. y = (1/two)x - 6
Here, the y-intercept = -6.
To determine the x-intercept, substitute y=0 and solve for x:
0 = (ane/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 whatsoever remaining answer choice that includes Three (E).

Hi GMATGuruNY
In the two cases below, did yous ready Y = 0 to find Y-intercept and X = 0 to find Ten-intercept ? Do nosotros demand to test both Y and 10 intercept to detrmine wether question is correct or not ? give thanks you and take a nice weekend

I. y = -x + 1
II. y = 2x + one
Each of these lines has a y-intercept betwixt -5 and five.
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).

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Re: A circle with center O and radius v is shown in the xy-airplane [#permalink]   twenty Jul 2018, 09:30

dave13 wrote:

Hi GMATGuruNY
In the two cases below, did yous set Y = 0 to find Y-intercept and 10 = 0 to find X-intercept ? Do we need to test both Y and 10 intercept to detrmine wether question is correct or not ? thank you and have a nice weekend

I. y = -x + one
2. y = 2x + one
Each of these lines has a y-intercept between -v and 5.
Thus, each line must pass through the interior of the circle.
Eliminate any answer choice that does not include both I and Two (A, B and D).

It is not necessary to find the x-intercepts for these ii lines.
Any line in the form y = mx + b has a y-intercept at (0, b).
Thus, each of the lines higher up has a y-intercept at (0, 1).
(0, 1) is within the interior of the circle.
Since each line has a y-intercept within the circle, each must pass through the interior of the circle and thus intersect the circumvolve at 2 points.
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Re: A circle with center O and radius 5 is shown in the xy-plane [#permalink]   26 Jun 2021, 07:36

VeritasKarishma wrote:

violetsplash wrote:

A circumvolve with center O and radius five is shown in the xy-aeroplane. Lines that intersect the circle in two points include which of the following ?

I. y = -10 +1
II. y = 2x + 1
III. y = (one/ii)ten - 6

a) I only
b) Two Only
c) I and 2 Only
d) I and III only
e) I, II and Three

Please see the attached picture

Co-ordinate geometry is all-time done past making diagrams of your own. On the ten axis, depict a circle with radius 5 and center at (0, 0). It will cut the x axis at 5 and -v and y axis at v and -5.

Now endeavour to plot the 3 given lines.

I. y = -ten +ane
Put x = 0, you get y = 1. So this line cuts the y axis at 1.
Put y = 0, yous get x = 1. And then this line cuts the ten centrality at 1.
This line, when extended on both sides will cut the circle at 2 singled-out points.

II. y = 2x + ane
Similarly, plot this line and you will come across that information technology volition cut the circle at two points also.

Three. y = (1/2)x - 6
This line cuts the y axis at -vi and x axis at 12. It is outside the circle and hence doesn't cutting the circle.

Hence right answer is (C)

Hey

So whenever center O is mentioned, we have to assume that eye O is origin?

Gagan

Re: A circle with center O and radius 5 is shown in the xy-plane [#permalink]

26 Jun 2021, 07:36

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Source: https://gmatclub.com/forum/a-circle-with-center-o-and-radius-5-is-shown-in-the-xy-plane-160576.html