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Circle C is in the xy-plane, what is the area of the circle?

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Circle C is in the xy-plane, what is the area of the circle?  [#permalink]

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New post 25 Aug 2010, 04:46
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Question Stats:

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Circle C is in the xy-plane, what is the area of the circle?

(1) Points (-2, 0) and (0,2) lie on the circle.
(2) The radius of the circle is equal to or less than 2^(1/2).
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Re: DS-Circle  [#permalink]

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New post 25 Aug 2010, 05:29
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jitendra wrote:
Circle C is in the xy-plane, what is the area of the circle?

(1) Points (-2, 0) and (0,2) lie on the circle.
(2) The radius of the circle is equal to or less than 2^(1/2).


\(area=\pi{r^2}\), so we should find the value of radius.

It would be better if you visualize this problem.

(1) Points (-2, 0) and (0,2) lie on the circle --> two points DO NOT define a circle (three points does), hence we can have numerous circles containing these two points, thus we can not find single numerical value of radius. Not sufficient.

Side note: if you put points (-2, 0) and (0,2) on XY-plane you can see that center of the circle must be on the line \(y=-x\) (the center of the circle must be equidistant from two pints given).

(2) The radius of the circle is equal to or less than \(\sqrt{2}\) --> \(r\leq{\sqrt{2}}\). Clearly insufficient.

(1)+(2) The distance between the 2 points given is \(d=\sqrt{2^2+2^2}=2\sqrt{2}\), so it's min length of diameter of the circle passing these points (diameter of a circle passing 2 points can not be less than the distance between these 2 points), thus half of \(2\sqrt{2}\) is min length of the radius of the circle --> \(r\geq{\sqrt{2}}\) but as from (2) \(r\leq{\sqrt{2}}\) then \(r=\sqrt{2}\) --> \(area=\pi{r^2}=2\pi\). Sufficient.

Answer: C.
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Re: Circle C is in the xy-plane, what is the area of the circle?  [#permalink]

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New post 28 Jun 2013, 22:46
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Adding to Bunnel's explanation, another way to arrive at why r >=\sqrt{2} is because the sum of any two sides of a triangle is greater than or equal to the third side. The triangle under consideration would be the one formed by the points (-2,0) , (0,2) and the center of the circle. Two sides of this triangle are equal to the radii of the circle. Thus, r+r >= distance between points (-2,0) , (0,2) = 2 \sqrt{2}.
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Re: Circle C is in the xy-plane, what is the area of the circle?  [#permalink]

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New post 08 Feb 2016, 18:01
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WilDThiNg wrote:
Bunuel wrote:
jitendra wrote:
Circle C is in the xy-plane, what is the area of the circle?

(1) Points (-2, 0) and (0,2) lie on the circle.
(2) The radius of the circle is equal to or less than 2^(1/2).


\(area=\pi{r^2}\), so we should find the value of radius.

It would be better if you visualize this problem.

(1) Points (-2, 0) and (0,2) lie on the circle --> two points DO NOT define a circle (three points does), hence we can have numerous circles containing these two points, thus we can not find single numerical value of radius. Not sufficient.



Hi:

Please could you help me understand and visualise the highlighted bit.

Thanks


What Bunuel means is that you can not 1 UNIQUE circle by just 2 points. As circle is a planar object, you need to have 3 distinct non collinear points to define a unique circle. Refer to the attached to see that both the circle pass through (-2,0) and (0,2). Clearly the radii of the 2 circles are different ---> giving you 2 different values of the areas. Thus this statement is NOT sufficient.

Attachment:
2016-02-08_19-59-00.jpg
2016-02-08_19-59-00.jpg [ 32.78 KiB | Viewed 3043 times ]


Hope this helps.
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Re: DS-Circle  [#permalink]

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New post 25 Aug 2010, 10:03
Good question...As Bunuel said its easier to visualise this and solve than just solving on paper!
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Re: DS-Circle  [#permalink]

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New post 05 May 2011, 22:44
a+b

distance = 2* 2^(1/2) and r <= 2^(1/2)

if r = 1, then diameter = 2 which is less than 2 * 2^(1/2).Not possible.
means r = 2^(1/2).

thus area = pi * [2^(1/2)]^2.

Hence C
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Re: Circle C is in the xy-plane, what is the area of the circle?  [#permalink]

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New post 27 Jun 2013, 22:30
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Re: DS-Circle  [#permalink]

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New post 06 Feb 2014, 20:51
Bunuel wrote:
jitendra wrote:

Side note: if you put points (-2, 0) and (0,2) on XY-plane you can see that center of the circle must be on the line \(y=-x\) (the center of the circle must be equidistant from two pints given).


Can you please elaborate on this point.
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Circle C is in the xy-plane, what is the area of the circle?  [#permalink]

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New post 17 Nov 2015, 19:03
neither statement alone is sufficient, yet, when we analyse both statements, we can clearly construct an isosceles triangle with base from 0;2 to 0;-2 or 4 and the angle opposite the base on the center of the circle. since there is only one value for the congruent sides to be equal on the center of the circle while the value of any 2 sides to be greater than the third, technically it is possible to identify the radius. Didn't go further... C
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Re: Circle C is in the xy-plane, what is the area of the circle?  [#permalink]

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New post 08 Feb 2016, 17:36
Bunuel wrote:
jitendra wrote:
Circle C is in the xy-plane, what is the area of the circle?

(1) Points (-2, 0) and (0,2) lie on the circle.
(2) The radius of the circle is equal to or less than 2^(1/2).


\(area=\pi{r^2}\), so we should find the value of radius.

It would be better if you visualize this problem.

(1) Points (-2, 0) and (0,2) lie on the circle --> two points DO NOT define a circle (three points does), hence we can have numerous circles containing these two points, thus we can not find single numerical value of radius. Not sufficient.



Hi:

Please could you help me understand and visualise the highlighted bit.

Thanks
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Re: Circle C is in the xy-plane, what is the area of the circle?  [#permalink]

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New post 08 Feb 2016, 18:06
Perfect - many thanks!
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Re: Circle C is in the xy-plane, what is the area of the circle?  [#permalink]

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Re: Circle C is in the xy-plane, what is the area of the circle?   [#permalink] 20 Jan 2019, 11:00
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