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Try this DS question

by gdk800 » Fri Nov 19, 2010 3:33 am
Hi All,

Please help me understand the following problem.

Question: Is the radius of the circle greater than 3?
1) (2,4) and (5,10) lie on the circle.
2) (2,4) and (4,1) lie on the circle.

My understanding is that the question asks us whether the radius of the circle greater than 3 or not? Thus if we are able to get a Yes or No from the option, it should suffice.
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by Rahul@gurome » Fri Nov 19, 2010 4:19 am
gdk800 wrote:Hi All,

Please help me understand the following problem.

Question: Is the radius of the circle greater than 3?
1) (2,4) and (5,10) lie on the circle.
2) (2,4) and (4,1) lie on the circle.

My understanding is that the question asks us whether the radius of the circle greater than 3 or not? Thus if we are able to get a Yes or No from the option, it should suffice.
Yes. Your understanding is correct.

Now to solve this problem, observe the fact that for any two points on circle, the greatest distance possible between them is equal to the diameter of the circle. Now, through two points infinite number of circles may be drawn. But the fact we just observed tells us that the smallest such circle will have the diameter equals to the distance between the points.

Statement 1: (2, 4) and (5, 10) lie on the circle.
Minimum possible diameter of the circle = Distance between (2, 4) and (5, 10)
= √[(5 - 2)² + (10 - 4)²]
= √[3² + 6²]

Clearly the minimum diameter is greater than 6.
Thus radius is always greater than 3.

Sufficient.

Statement 1: (2, 4) and (4, 1) lie on the circle.
Minimum possible diameter of the circle = Distance between (2, 4) and (4, 1)
= √[(4 - 2)² + (1 - 4)²]
= √[2² + 3²]
= √13

As √13 < 6, the minimum diameter is less than 6.
Thus radius is may or may not be greater than 3.

Not sufficient.

The correct answer is A.
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by gdk800 » Fri Nov 19, 2010 7:44 am
gdk800 wrote:
Hi All,

Please help me understand the following problem.

Question: Is the radius of the circle greater than 3?
1) (2,4) and (5,10) lie on the circle.
2) (2,4) and (4,1) lie on the circle.

My understanding is that the question asks us whether the radius of the circle greater than 3 or not? Thus if we are able to get a Yes or No from the option, it should suffice.
Yes. Your understanding is correct.

Now to solve this problem, observe the fact that for any two points on circle, the greatest distance possible between them is equal to the diameter of the circle. Now, through two points infinite number of circles may be drawn. But the fact we just observed tells us that the smallest such circle will have the diameter equals to the distance between the points.

Statement 1: (2, 4) and (5, 10) lie on the circle.
Minimum possible diameter of the circle = Distance between (2, 4) and (5, 10)
= √[(5 - 2)² + (10 - 4)²]
= √[3² + 6²]

Clearly the minimum diameter is greater than 6.
Thus radius is always greater than 3.

Sufficient.

Statement 1: (2, 4) and (4, 1) lie on the circle.
Minimum possible diameter of the circle = Distance between (2, 4) and (4, 1)
= √[(4 - 2)² + (1 - 4)²]
= √[2² + 3²]
= √13

As √13 < 6, the minimum diameter is less than 6.
Thus radius is may or may not be greater than 3.

Not sufficient.

The correct answer is A.
_________________


Firstly, thanks Rahul for posting the answer i was trying to get to because the following explains the reason i put this post for.

As √13 < 6, the minimum diameter is less than 6.
Thus radius is may or may not be greater than 3.


As √13 < 6, the minimum diameter is less than 6.
this means the radius is also less than 3 coz if Diameter < 6 than Diameter/2 (= Radius ) < 6/2 and thus it answers our question that the radius is NOT greater than 3. This is the reason i marked D and not A, which is the OA....


Kindly clarify, I will be much thankful
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by goyalsau » Fri Nov 19, 2010 10:59 am
Rahul@gurome wrote: Now to solve this problem, observe the fact that for any two points on circle, the greatest distance possible between them is equal to the diameter of the circle. Now, through two points infinite number of circles may be drawn. But the fact we just observed tells us that the smallest such circle will have the diameter equals to the distance between the points.

Rahul i thought the answer of this question is C, Will make one equation from the I statement as the radius of the circle will be equal from all the points ( x, y ) is the center of the circle.

( x - 2 ) ^ 2 + ( y - 4 ) ^ 2 = ( x - 5 ) ^ 2 + ( y - 10) ^ 2

Like the same way will have II equation from II statement,

Never thought about the concept of the greatest distance between points on the circle,

My question is If on the coordinate plane , coordinates of 2 points on the circle are given . Can we be sure of the radius of the circle , Or we need at least 3 coordinate points to find the radius of the circle.
Saurabh Goyal
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by Rahul@gurome » Fri Nov 19, 2010 11:22 am
gdk800 wrote:
As √13 < 6, the minimum diameter is less than 6.
this means the radius is also less than 3 coz if Diameter < 6 than Diameter/2 (= Radius ) < 6/2 and thus it answers our question that the radius is NOT greater than 3. This is the reason i marked D and not A, which is the OA....


Kindly clarify, I will be much thankful
Please note that we are talking about minimum possible diameter for the circle which will pass through the given points. This is not the diameter of the circle! This means the actual diameter of the circle may be 4 or 6 or 12 or anything but greater than √13. Thus radius may be 2 or 3 or 5 or 10 or anything greater than (√13/2).
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by Rahul@gurome » Fri Nov 19, 2010 11:25 am
goyalsau wrote:My question is If on the coordinate plane , coordinates of 2 points on the circle are given . Can we be sure of the radius of the circle , Or we need at least 3 coordinate points to find the radius of the circle.
No. From only two points we cannot determine the radius of the circle, we need at least three points. But as I mentioned earlier, we can definitely determine the minimum possible radius of the circles that will go through the points.
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by goyalsau » Fri Nov 19, 2010 7:20 pm
Rahul@gurome wrote:
goyalsau wrote:My question is If on the coordinate plane , coordinates of 2 points on the circle are given . Can we be sure of the radius of the circle , Or we need at least 3 coordinate points to find the radius of the circle.
No. From only two points we cannot determine the radius of the circle, we need at least three points. But as I mentioned earlier, we can definitely determine the minimum possible radius of the circles that will go through the points.
Thanks, Rahul....
Saurabh Goyal
[email protected]
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