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

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anik89
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The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink] New post   24 Nov 2014, 22:53
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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
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A
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BrentGMATPrepNow
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The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink] New post   Updated on: 16 Feb 2021, 17:47
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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


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

Answer:
Show Spoiler
A


Cheers,
Brent

Originally posted by BrentGMATPrepNow on 25 Oct 2017, 10:19.
Last edited by BrentGMATPrepNow on 16 Feb 2021, 17:47, edited 2 times in total.
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Re: The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink] New post   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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Re: The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink] New post   24 Nov 2014, 23:21
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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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Re: The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink] New post   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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manpreetsingh86
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Re: The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink] New post   28 Nov 2014, 08:53
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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
sol.jpg [ 7.82 KiB | Viewed 27884 times ]

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Re: The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink] New post   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] New post   Updated on: 06 Jul 2017, 17:06
The radius can be 5,6,7 or 8. All these cases satisfy the two given conditions.

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] New post   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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The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink] New post   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] New post   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] New post   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 :)
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Re: The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink] New post   11 Jan 2019, 20:09
Bunuel,
Can we tag this question to Co-ordinate Geometry.

Thanks
Probus
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Re: The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink] New post   12 Jan 2019, 03:30
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Probus wrote:
Bunuel,
Can we tag this question to Co-ordinate Geometry.

Thanks
Probus

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Done. Thank you.
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Re: The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink] New post   13 Jan 2020, 09:21
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



(5, 7) is outside the circle & r, is an integer -->>mean maximum radius is '8'
(1, -2) is inside the circle -->>mean minimum radius is 5

There are only 4 Integer from 5 to 8, ie 5,6,7,8 (possible values of 'r'. Thus answer is 4, ie A
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The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink] New post   02 May 2021, 14:24
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Bunuel

can you please help me on this question?

Can you confirm if a point "inside" a circle can never be on the edge of a circle? For example, for a square that is inscribed in a circle, the corner points would not be considered inside the circle?
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Re: The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink] New post   02 May 2021, 22:57
From which Official Guide does this question originate?

I can’t seem to find it in any of my Guides.

Good question, just curious.

Posted from my mobile device
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Re: The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink] New post   15 May 2021, 15:48
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ef1988 wrote:
Bunuel

can you please help me on this question?

Can you confirm if a point "inside" a circle can never be on the edge of a circle? For example, for a square that is inscribed in a circle, the corner points would not be considered inside the circle?


I thought that "being inside the circle" means that the can be in the edge of the circle, that's why I considered that R could take 5 values: 4,5,6,7,8.

Having this question wrong, does it mean that "inside the circle" means that being in the edge of the circle is not "inside"???

Best regards!
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Re: The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink] New post   16 May 2021, 04:15
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Also wonder whether its always the case that "inside the circle" can not mean "on the border" of the circle. In vs On the circle?

Posted from my mobile device
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Re: The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink] New post   03 Jul 2021, 07:55
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Bambi2021 wrote:
Also wonder whether its always the case that "inside the circle" can not mean "on the border" of the circle. In vs On the circle?

Posted from my mobile device


Hey Bambi2021, inside the circle means it is inside the circle. It can be lying along the borders but it can not be on the border.
Whereas 'On' the circle means it can only lie on the circumference of the circle. Hope this helps :)
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Re: The center of a circle is (5, -2). (5, 7) is outside the circle, [#permalink]
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