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

Question

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

BrentGMATPrepNow · Accepted Answer

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)
 https://i.imgur.com/VSZinV5.png

(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...
 https://i.imgur.com/MkWN90m.png

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  
 https://i.imgur.com/Hg6DUTe.png

(5, 7) is OUTSIDE the circle  
The distance from (5, 7) to the circle's center (5, -2) is 9 units...
 https://i.imgur.com/hcn2E0o.png 
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:  A

Cheers,
Brent

manpreetsingh86 · Answer

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.

Temurkhon · Answer

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

smartyguy · Answer

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

manpreetsingh86 · Answer

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

RussianDude · Answer

r must be greater than 4 and smaller than 9, hence r=5,6,7 or 8. Answer A

sasyaharry · Answer

The radius can be 5,6,7 or 8. All these cases satisfy the two given conditions.

pushpitkc · Answer

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)

dave13 · Answer

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

abhimahna · Answer

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?

dave13 · Answer

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

Probus · Answer

Bunuel ,
Can we tag this question to Co-ordinate Geometry.

Thanks
Probus

Bunuel · Answer

_________________
Done. Thank you.

udaypratapsingh99 · Answer

(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

ef1988 · Answer

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?

Fdambro294 · Answer

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

JoaquinCerda · Answer

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!

Bambi2021 · Answer

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

errorlogger · Answer

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

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards