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In the coordinate system, the center of a circle lies at (2, 3)

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In the coordinate system, the center of a circle lies at (2, 3)  [#permalink]

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New post 19 Mar 2018, 19:36
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In the coordinate system, the center of a circle lies at (2, 3). If point A with coordinates (-1, 7) does not lie outside the circle, which of the following points must lie inside the circle?

I. (0, 7)
II. (5, -1)
III. (-2, 7)

A) I only
B) II only
C) III only
D) I and II only
E) None of the above

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Re: In the coordinate system, the center of a circle lies at (2, 3)  [#permalink]

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New post 19 Mar 2018, 21:02
saswata4s wrote:
In the coordinate system, the center of a circle lies at (2, 3). If point A with coordinates (-1, 7) does not lie outside the circle, which of the following points must lie inside the circle?

I. (0, 7)
II. (5, -1)
III. (-2, 7)

A) I only
B) II only
C) III only
D) I and II only
E) None of the above


The maximum distance which Point A can lie from centre = 5 (Its the distance from centre of circle to Point A).
(5,-1) & (-2,7) lies at the circle as their distance from Centre of Circle is 5. Distance from (0,7) to (2,3) is less than 5. Hence, (0,7) lies inside the circle.

Option A.

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Re: In the coordinate system, the center of a circle lies at (2, 3)  [#permalink]

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New post 19 Mar 2018, 23:52
r <=5 since the point must be on or inside the circle.
Option I gives R < 5. If R = 5 we do not know for certain if the point is inside the circle, hence option II is out. Option III is anyways greater than 5.

A.
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Re: In the coordinate system, the center of a circle lies at (2, 3)  [#permalink]

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New post 20 Mar 2018, 02:08
Origin, O - (2,3)
A (-1,7) - Does not lie outside the circle, implies it is either on the circle or inside the circle.

Distance between (2,3) and (-1,7) = 5
So, radius of the circle <= 5

Q) Must lie inside the circle, implies NEITHER outside the circle NOR on the circle.

D < 5 ===> D!>=5

Point 1 - Distance between (2,3) and (0,7) ===> Less than 5
Point 2 - Distance between (2,3) and (5,-1) ===> Equals to 5
Point 3 - Distance between (2,3) and (-2,7) ===> More than 5

Answer is Only I.

Option A.

saswata4s wrote:
In the coordinate system, the center of a circle lies at (2, 3). If point A with coordinates (-1, 7) does not lie outside the circle, which of the following points must lie inside the circle?

I. (0, 7)
II. (5, -1)
III. (-2, 7)

A) I only
B) II only
C) III only
D) I and II only
E) None of the above
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Re: In the coordinate system, the center of a circle lies at (2, 3)  [#permalink]

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New post 25 Mar 2018, 10:51
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1

Solution



If the point \(A (-1,7\)) does not lie outside the circle, then it can lie either on the circle or inside the circle as shown in the diagram.


Image

Let us denote the distance between the point \(A (-1,7)\) and centre of the circle \((2,3)\) by \(D\).
    D= \(\sqrt{(2-(-1))^2+(3-7)^2}\)
    D= 5

Now, when the distance of point A is 5, then we are uncertain whether it lies on the circle or inside the circle.

However, if the distance of any point from centre is less than 5 then we can infer, that the point is certainly inside the circle.

Let us find the distance of all the points given in options.

1- (0,7)

Distance= \(\sqrt{(2-0)^2+(3-7)^2}\)
Distance= \(\sqrt{20}\), which is less than \(5\).

Hence (0,7) lies inside the circle.

2- (5, -1)

Distance= \(\sqrt{(2-5)^2+(3-(-1))^2}\)
Distance= \(5\)

3- (-2, 7)

Distance= \(\sqrt{(2-(-2))^2+(3-7)^2}\)
Distance= \(\sqrt{32}\), which is greater than \(5\).

Answer: A
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Re: In the coordinate system, the center of a circle lies at (2, 3) &nbs [#permalink] 25 Mar 2018, 10:51
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