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# Curve Question

Author Message
Manager
Joined: 12 Feb 2006
Posts: 115

Kudos [?]: 80 [0], given: 0

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22 Feb 2007, 21:48
00:00

Difficulty:

(N/A)

Question Stats:

25% (00:00) correct 75% (01:04) wrong based on 8 sessions

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This one got me. Any help would be appreciated.

How many points of intersection does curve x^2 + y^2 = 4 have with x + y = 2?

0
1
2
3
4

Kudos [?]: 80 [0], given: 0

Senior Manager
Joined: 12 Mar 2006
Posts: 363

Kudos [?]: 75 [0], given: 3

Schools: Kellogg School of Management

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22 Feb 2007, 22:14
bz9 wrote:
This one got me. Any help would be appreciated.

How many points of intersection does curve x^2 + y^2 = 4 have with x + y = 2?

0
1
2
3
4

we can rule out D & E since a line can instersect a circle only in 2 pts....

The points of intersection will satisfy both equations x^2 + y^2 = 4 and x + y = 2

so x = y - 2

=> (y-2)^2 + y^2 = 0
2y^2 - 4y + 4 = 0
y^2 - 2y + 2 = 0
solve quadratic eq to get values of x,y
if discriminant is >0 then we have 2pts of intersection
if discriminant is =0 then we have 1pt of intersection (line is a tangent to the circle)
if discriminant is <0 then we have 0pts of intersection (line does not intersect the circle)

in this case discriminant = 4 - 8 = -4 so ans is 0 (A)

Kudos [?]: 75 [0], given: 3

Intern
Joined: 16 Aug 2006
Posts: 17

Kudos [?]: [0], given: 0

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22 Feb 2007, 22:43
here's another way, i realized (after a while) that x^2 +y^2 =4 is the equation for a circle with radius=2 and centered at the origin.

http://www.mathwarehouse.com/geometry/c ... circle.php

x+y=2 is the equation of a line and it won't touch the circle more than twice so you can eliminate the last two. once you note that one is a circle and the other a line i think the next easiest step is to draw a fast coordinate plane in the notepad and see if the line passes through or is tangent to the circle, since the equation for this particular line and circle are simple. The other way is to solve both for y, knowing that the maximum number of intersections is two.

y1=2-x

y2=SQRT(4-x^2)

At x=0, y1=y2=2.
At x=2, y1=y2=0.

So the line passes through the circle twice, at (0,2) and at (2,0).

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Manager
Joined: 12 Sep 2006
Posts: 81

Kudos [?]: 18 [0], given: 0

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23 Feb 2007, 07:28
prude_sb wrote:
The points of intersection will satisfy both equations x^2 + y^2 = 4 and x + y = 2

so x = y - 2

I think it should be x=2-y

Also , The equation of the given circle is x^2+Y^=4

prude_sb wrote:
(y-2)^2 + y^2 = 0

What do you say?

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SVP
Joined: 01 May 2006
Posts: 1794

Kudos [?]: 167 [0], given: 0

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23 Feb 2007, 07:32
sgoll wrote:
prude_sb wrote:
The points of intersection will satisfy both equations x^2 + y^2 = 4 and x + y = 2

so x = y - 2

I think it should be x=2-y

Also , The equation of the given circle is x^2+Y^=4

prude_sb wrote:
(y-2)^2 + y^2 = 0

What do you say?

Well, it's not to me ... But I'm passing here

Notice that : (2-y)^2 = ((-1)*(y-2))^2 = (-1)^2*(y-2)^2 = (y-2)^2

For the other part.... U are right... 4 is missing

Kudos [?]: 167 [0], given: 0

Manager
Joined: 12 Sep 2006
Posts: 81

Kudos [?]: 18 [0], given: 0

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23 Feb 2007, 08:00
Fig wrote:
sgoll wrote:
prude_sb wrote:
The points of intersection will satisfy both equations x^2 + y^2 = 4 and x + y = 2

so x = y - 2

I think it should be x=2-y

Also , The equation of the given circle is x^2+Y^=4

prude_sb wrote:
(y-2)^2 + y^2 = 0

What do you say?

Well, it's not to me ... But I'm passing here

Notice that : (2-y)^2 = ((-1)*(y-2))^2 = (-1)^2*(y-2)^2 = (y-2)^2

For the other part.... U are right... 4 is missing

I agree (y-2)^=(2-y)^2
Further, the eqn when equals 4, it results to answer to C: 2
Explanation
substituting x=2-y in the eqn of the Circle we get
(2-y)^2+y^2=4
Solving for y we get y(y-2)=0
=> either y=0 or y=2 giving two points of intersection (0,2) and (2,0)

Kudos [?]: 18 [0], given: 0

SVP
Joined: 01 May 2006
Posts: 1794

Kudos [?]: 167 [0], given: 0

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23 Feb 2007, 08:11
sgoll wrote:
Fig wrote:
sgoll wrote:
prude_sb wrote:
The points of intersection will satisfy both equations x^2 + y^2 = 4 and x + y = 2

so x = y - 2

I think it should be x=2-y

Also , The equation of the given circle is x^2+Y^=4

prude_sb wrote:
(y-2)^2 + y^2 = 0

What do you say?

Well, it's not to me ... But I'm passing here

Notice that : (2-y)^2 = ((-1)*(y-2))^2 = (-1)^2*(y-2)^2 = (y-2)^2

For the other part.... U are right... 4 is missing

I agree (y-2)^=(2-y)^2
Further, the eqn when equals 4, it results to answer to C: 2
Explanation
substituting x=2-y in the eqn of the Circle we get
(2-y)^2+y^2=4
Solving for y we get y(y-2)=0
=> either y=0 or y=2 giving two points of intersection (0,2) and (2,0)

Yes ... Absolultly

Kudos [?]: 167 [0], given: 0

CEO
Joined: 21 Jan 2007
Posts: 2734

Kudos [?]: 1050 [0], given: 4

Location: New York City

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05 Nov 2007, 09:26
OA is C.

Two roots yields two coordinates ===> 2 points of intersection.

Kudos [?]: 1050 [0], given: 4

Intern
Joined: 17 Sep 2011
Posts: 13

Kudos [?]: 2 [1], given: 9

Schools: LBS '14
GMAT Date: 10-24-2011

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22 Sep 2011, 07:10
1
KUDOS
bz9 wrote:
This one got me. Any help would be appreciated.

How many points of intersection does curve x^2 + y^2 = 4 have with x + y = 2?

0
1
2
3
4

How'bout my solution?
In x + y = 2, we see y-intercept =(0,2), x-intercept =(2,0)
In the curve that is centered by the origin, we know it passes (0,2), (-2,0), (0,-2), and (2,0).
So, the line and the curve meet at two points.

Kudos [?]: 2 [1], given: 9

Manager
Status: Essaying
Joined: 27 May 2010
Posts: 146

Kudos [?]: 9 [0], given: 8

Location: Ghana
Concentration: Finance, Finance
Schools: Cambridge
GMAT 1: 690 Q47 V37
GPA: 3.9
WE: Accounting (Education)

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23 Sep 2011, 02:55
Got two points sub equ 2 into 1 and determine with b^2 - 4ac

Kudos [?]: 9 [0], given: 8

Director
Joined: 01 Feb 2011
Posts: 726

Kudos [?]: 144 [0], given: 42

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25 Sep 2011, 13:52
x^2 + y^2 = 4 is a circle with radius 2.
x+y = 2 is a straight line , with slope -1 , x intercept = 2 , y intercept = 2.
=> point of intersection = (2,0) and (0,2) = 2 points

Kudos [?]: 144 [0], given: 42

Current Student
Joined: 08 Jan 2009
Posts: 323

Kudos [?]: 159 [0], given: 7

GMAT 1: 770 Q50 V46

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26 Sep 2011, 19:42
No need to get too bogged down on the maths here. We have a circle and a line, so we are either going to have zero intersections, one intersection or two intersections.

Sketch out the circle, sketch the line. The line intersects x and y at 2, as does the circle. Two intersections.

Kudos [?]: 159 [0], given: 7

Manager
Joined: 09 Jul 2009
Posts: 52

Kudos [?]: 86 [0], given: 27

Location: Bangalore
Schools: ISB

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27 Sep 2011, 12:45
me too first got A then C
_________________

The only way of finding the limits of the possible is by going beyond them into the impossible.

Kudos [?]: 86 [0], given: 27

Current Student
Joined: 21 Aug 2010
Posts: 203

Kudos [?]: 17 [0], given: 28

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29 Sep 2011, 00:29
It was a simple one.

A circle and a line intersection. Either 0, 1 or 2 points of intersection. Rest has been explained very well by others.

BR
Mandeep

Kudos [?]: 17 [0], given: 28

Re: Curve Question   [#permalink] 29 Sep 2011, 00:29
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