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

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

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26 Sep 2013, 07:25
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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 + 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
[Reveal] Spoiler: OA

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Circle.jpg [ 8.5 KiB | Viewed 21278 times ]

Last edited by Bunuel on 01 Nov 2014, 07:36, edited 3 times in total.
RENAMED THE TOPIC.

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

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26 Sep 2013, 08:08
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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

Check the diagram below:
Attachment:

graph.png [ 9.35 KiB | Viewed 21260 times ]
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Re: A circle with center O and radius 5 is shown in the xy-plane [#permalink]

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26 Sep 2013, 09:16
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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
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Re: A circle with center O and radius 5 is shown in the xy-plane [#permalink]

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26 Sep 2013, 09:41
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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

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

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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

Check the diagram below:
Attachment:
graph.png

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

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

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09 Jun 2014, 21:58
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Expert's post
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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

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.

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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 17321 [12], given: 232 Current Student Joined: 26 Aug 2014 Posts: 827 Kudos [?]: 163 [3], given: 98 Concentration: Marketing GPA: 3.4 Re: A circle with center O and radius 5 is shown in the xy-plane [#permalink] ### Show Tags 01 Nov 2014, 07:28 3 This post received KUDOS 1 This post was BOOKMARKED 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. Kudos [?]: 163 [3], given: 98 Intern Joined: 21 Feb 2015 Posts: 27 Kudos [?]: 8 [1], given: 23 A circle with center O and radius 5 is shown in the xy-plane [#permalink] ### Show Tags 02 Apr 2015, 18:48 1 This post received KUDOS 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? Kudos [?]: 8 [1], given: 23 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7668 Kudos [?]: 17321 [1], given: 232 Location: Pune, India Re: A circle with center O and radius 5 is shown in the xy-plane [#permalink] ### Show Tags 02 Apr 2015, 19:29 1 This post received KUDOS Expert's post 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 My Blog Get started with Veritas Prep GMAT On Demand for$199

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

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03 Apr 2015, 06:21
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?

Last edited by Rookie124 on 03 Apr 2015, 06:39, edited 1 time in total.

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

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02 May 2015, 02:33
not easy at all

find out the points a line intersect the two axis x and y. this means pick x=0 and y=0 for each of equation
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Re: A circle with center O and radius 5 is shown in the xy-plane [#permalink]

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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.

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

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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?

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

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

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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,

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

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11 Jul 2016, 22:22
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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.
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A circle with center O and radius 5 is shown in the xy-plane [#permalink]

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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

Check the diagram below:
Attachment:
graph.png

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.

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

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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

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.

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.

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Re: A circle with center O and radius 5 is shown in the xy-plane   [#permalink] 01 Apr 2017, 13:25

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