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# The center of a circle is (5, -2). (5, 7) is outside the circle,

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Intern
Joined: 08 May 2014
Posts: 8
The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink]

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24 Nov 2014, 22:53
4
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45% (medium)

Question Stats:

66% (01:32) correct 34% (01:38) wrong based on 389 sessions

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The center of a circle is (5, -2). (5, 7) is outside the circle, and (1, -2) is inside the circle. If the radius, r, is an integer, how many possible values are there for r?

(A) 4
(B) 5
(C) 6
(D) 7
(E) 8
Director
Joined: 23 Jan 2013
Posts: 596
Schools: Cambridge'16
Re: The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink]

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24 Nov 2014, 23:21
Idea is that possible radius is between two points: (1,-2) and (5,7)

5-1=4, meaning that point inside the circle is 4 numbers out of center (5,-2),
we should count it in Y axe in (5,7) direction, so -2+4=2 and radius is between 2 and 7.

7-2 =5 values of R, but 7 is out and maximal possible is 4

A
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Posts: 277
Re: The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink]

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25 Nov 2014, 02:52
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anik89 wrote:
The center of a circle is (5, -2). (5, 7) is outside the circle, and (1, -2) is inside the circle. If the radius, r, is an
integer, how many possible values are there for r?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

distance between the center of the circle (5,-2) and (1,-2) = $$\sqrt{(5-1)^2 + (-2+2)^2}$$ $$= 4$$

distance between the center of the circle (5,-2) and (5,7) = $$\sqrt{(5-5)^2 + (7+2)^2}$$ $$= 9$$

now, since point (1,-2) lies inside the circle, therefore radius of the circle will be greater than 4. and also since point (5,7) lies outside the circle, therefore radius of the circle will be less than 9

i.e. 4<r<9
now since r is an integer, therefore possible values of r are 5,6,7 and 8. hence answer should be A.
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Joined: 27 Nov 2014
Posts: 61
Re: The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink]

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28 Nov 2014, 00:47
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The center of a circle is (5, -2). (5, 7) is outside the circle, and (1, -2) is inside the circle. If the radius, r, is an
integer, how many possible values are there for r?

does inside the circle means not on the circle ?

if no then (5,7) if lies on the circle then we should consider that point also.

than also total is 5

Please clarify and correct where m i wrong ?

Regards
SG
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Re: The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink]

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28 Nov 2014, 08:53
1
KUDOS
smartyguy wrote:
The center of a circle is (5, -2). (5, 7) is outside the circle, and (1, -2) is inside the circle. If the radius, r, is an
integer, how many possible values are there for r?

does inside the circle means not on the circle ?

if no then (5,7) if lies on the circle then we should consider that point also.

than also total is 5

Please clarify and correct where m i wrong ?

Regards
SG

hi, the situation mentioned in the question can be visualized in the following manner. i hope it helps.
Attachments

sol.jpg [ 7.82 KiB | Viewed 4214 times ]

Intern
Joined: 11 Apr 2015
Posts: 35
Location: Germany
Concentration: General Management, Entrepreneurship
GPA: 3.1
WE: Project Management (Energy and Utilities)
Re: The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink]

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30 Jun 2016, 10:19
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r must be greater than 4 and smaller than 9, hence r=5,6,7 or 8. Answer A
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Re: The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink]

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Updated on: 06 Jul 2017, 17:06
The radius can be 5,6,7 or 8. All these cases satisfy the two given conditions.
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Originally posted by sasyaharry on 06 Jul 2017, 13:31.
Last edited by sasyaharry on 06 Jul 2017, 17:06, edited 1 time in total.
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Re: The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink]

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06 Jul 2017, 13:51
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Given data :
R, the radius of the circle has to be an integer
Center of the circle (5,-2)
Point inside the circle (1,-2)
Point outside the circle (5,7)

Formula used :
Distance between 2 points (x1,y1) and (x2,y2) is $$\sqrt{(x1-x2)^2 + (y1-y2)^2}$$

Since a point inside the circle is (1,-2).
The distance from the center of the circle is $$\sqrt{(5-1)^2 + (-2+2)^2}$$(which is equal to 4)
The point outside the circle is (5,7)
The distance from the center of the circle is $$\sqrt{(5-5)^2 + (7+2)^2}$$(which is equal to 9)

Since we have found out the range of the radius which is 4 < R < 9
We have four values for R(5,6,7,8) (Option A)
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Joined: 12 Sep 2015
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The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink]

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Updated on: 16 Apr 2018, 12:23
Expert's post
Top Contributor
anik89 wrote:
The center of a circle is (5, -2). (5, 7) is outside the circle, and (1, -2) is inside the circle. If the radius, r, is an
integer, how many possible values are there for r?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

The center of a circle is (5, -2)
As we can see from the image below, there are infinitely many circles that have their center at (5, -2)

(1, -2) is INSIDE the circle
We can quickly determine that the distance from (1, -2) to the circle's center (5, -2) is 4 units...

So a circle with a radius of 4 would PASS THROUGH the point (1, -2).
Since (1, -2) is INSIDE the circle, we know that the radius of the circle must be GREATER THAN 4

(5, 7) is OUTSIDE the circle
The distance from (5, 7) to the circle's center (5, -2) is 9 units...

So a circle with a radius of 9 would PASS THROUGH the point (5, 7).
Since (5, 7) is OUTSIDE the circle, we know that the radius of the circle must be LESS THAN 9

So, we now have lower and upper limits of the radius.
The radius of the circle is GREATER THAN 4 and LESS THAN 9
In other words, 4 < r < 9

Since the radius, r, is an INTEGER, there are only four possible values of r.
r can equal 5, 6, 7, or 8

Cheers,
Brent
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Originally posted by GMATPrepNow on 25 Oct 2017, 10:19.
Last edited by GMATPrepNow on 16 Apr 2018, 12:23, edited 1 time in total.
Director
Joined: 09 Mar 2016
Posts: 533
The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink]

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31 Mar 2018, 07:21
pushpitkc wrote:
Given data :
R, the radius of the circle has to be an integer
Center of the circle (5,-2)
Point inside the circle (1,-2)
Point outside the circle (5,7)

Formula used :
Distance between 2 points (x1,y1) and (x2,y2) is $$\sqrt{(x1-x2)^2 + (y1-y2)^2}$$

Since a point inside the circle is (1,-2).
The distance from the center of the circle is $$\sqrt{(5-1)^2 + (-2+2)^2}$$(which is equal to 4)
The point outside the circle is (5,7)
The distance from the center of the circle is $$\sqrt{(5-5)^2 + (7+2)^2}$$(which is equal to 9)

Since we have found out the range of the radius which is 4 < R < 9
We have four values for R(5,6,7,8) (Option A)

Hi pushpitkc,
i have one technical question
$$\sqrt{(5-1)^2 + (-2+2)^2}$$(which is equal to 4)

can you expand it how you got 4 ... after taking square root i get this (5-1) + (2+2) which is 8
thank you
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The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink]

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31 Mar 2018, 07:55
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dave13 wrote:
Hi pushpitkc,
i have one technical question
$$\sqrt{(5-1)^2 + (-2+2)^2}$$(which is equal to 4)

can you expand it how you got 4 ... after taking square root i get this (5-1) + (2+2) which is 8
thank you

Hey dave13 ,

$$\sqrt{(a)^2 + (b)^2}$$ is not equal to $$a + b$$

You should always solve the equation first (meaning add $$a^2$$ and $$b^2$$) and then take the square root.

Does that make sense?
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Re: The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink]

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31 Mar 2018, 08:15
abhimahna wrote:
dave13 wrote:
Hi pushpitkc,
i have one technical question
$$\sqrt{(5-1)^2 + (-2+2)^2}$$(which is equal to 4)

can you expand it how you got 4 ... after taking square root i get this (5-1) + (2+2) which is 8
thank you

Hey dave13 ,

$$\sqrt{(a)^2 + (b)^2}$$ is not equal to $$a + b$$

You should always solve the equation first (meaning add $$a^2$$ and $$b^2$$) and then take the square root.

Does that make sense?

many thanks abhimahna yes it make totally perfect sence

$$\sqrt{(5-1)^2 + (-2+2)^2}$$(which is equal to 4)

so i get $$\sqrt{(4)^2 + (0)^2}$$

$$4 + 0$$ = 4 fantastic
Re: The center of a circle is (5, -2). (5, 7) is outside the circle,   [#permalink] 31 Mar 2018, 08:15
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