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What are the coordinates of point A?

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What are the coordinates of point A?  [#permalink]

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New post 13 Jun 2013, 08:21
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What are the coordinates of point A?

(1) A is 2 units away from (3,4)
(2) A is 3 units away from (0,0)
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What are the coordinates of point A?  [#permalink]

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New post 13 Jun 2013, 09:28
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arpanpatnaik wrote:
sps1604 wrote:
What are the coordinates of point A?
1) A is 2 units away from (3,4)
2) A is 3 units away from (0,0)

How to solve this question?


The answer is [C]. The approach would be using the statements given.

We are to find the co-ordinates of point A. Hence they have to be a unique set of {x,y}.
Now Statement 1 states, A is 2 units away from (3,4). So basically all points at a distance of 2 units from (3,4) can be A. Hence the locus of Point A is a circle with the equation: \((x-3)^2 + (y-4)^2 = 9\). Clearly Statement 1 alone is not sufficient.

Now Statement 2 states A is 3 units away from (0,0). Similarly it can said that A can be a circle with its center on (0,0) with a radius of 2. Equation: \(x^2 + y^2 = 4\). Again its insufficient alone.

As we combine them we know that two circles can intersect at two points of maybe just touch at one! A would be plausible only if the circles meet at one point. Now if the circles met at one point the distance between the centers = sum of the radii. Distance between centers = 5 units and clearly sum of radii is 5! Hence we can tell for sure that the circles touch at one point which is A. Therefore the answer is [C].

Hope it helps! :)

Regards,
A


Just to illustrate:

Image

Hope it helps.

Attachment:
Distance.png
Distance.png [ 11.07 KiB | Viewed 6091 times ]

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Re: What are the coordinates of point A?  [#permalink]

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New post 13 Jun 2013, 09:12
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sps1604 wrote:
What are the coordinates of point A?
1) A is 2 units away from (3,4)
2) A is 3 units away from (0,0)

How to solve this question?


The answer is [C]. The approach would be using the statements given.

We are to find the co-ordinates of point A. Hence they have to be a unique set of {x,y}.
Now Statement 1 states, A is 2 units away from (3,4). So basically all points at a distance of 2 units from (3,4) can be A. Hence the locus of Point A is a circle with the equation: \((x-3)^2 + (y-4)^2 = 9\). Clearly Statement 1 alone is not sufficient.

Now Statement 2 states A is 3 units away from (0,0). Similarly it can said that A can be a circle with its center on (0,0) with a radius of 2. Equation: \(x^2 + y^2 = 4\). Again its insufficient alone.

As we combine them we know that two circles can intersect at two points of maybe just touch at one! A would be plausible only if the circles meet at one point. Now if the circles met at one point the distance between the centers = sum of the radii. Distance between centers = 5 units and clearly sum of radii is 5! Hence we can tell for sure that the circles touch at one point which is A. Therefore the answer is [C].

Hope it helps! :)

Regards,
A
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Re: What are the coordinates of point A?  [#permalink]

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New post 13 Jun 2013, 10:04
sps1604 wrote:
What are the coordinates of point A?

(1) A is 2 units away from (3,4)
(2) A is 3 units away from (0,0)


Clearly (1) and (2) are alone not sufficient to answer the question, as the point A can be in any direction of the given points.

Combining (1) and (2),
Suppose, B = (3,4) and C = (0,0) , You need to find co-ordinates of A and been given distances of the point from B and C.

You can pin-point A, only if it lies on the line BC.

Otherwise, for any combination of distances you will always have two positions of A, i.e above(left) or below(right) line BC. (Visualize ABC as a triangle with BC as base, for same values of AB and AC you can always construct two triangles).

quickly check: length of BC =\(\sqrt{3^2 + 4^2}\) = 5
AB = 2 , AC = 3

Since AB + AC = BC , A lies on BC, hence you can determine the co-ordinates of A.


Answer C.
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Re: What are the coordinates of point A?  [#permalink]

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New post 13 Jun 2013, 20:04
arpanpatnaik wrote:
sps1604 wrote:
What are the coordinates of point A?
1) A is 2 units away from (3,4)
2) A is 3 units away from (0,0)

How to solve this question?


The answer is [C]. The approach would be using the statements given.

We are to find the co-ordinates of point A. Hence they have to be a unique set of {x,y}.
Now Statement 1 states, A is 2 units away from (3,4). So basically all points at a distance of 2 units from (3,4) can be A. Hence the locus of Point A is a circle with the equation: \((x-3)^2 + (y-4)^2 = 9\). Clearly Statement 1 alone is not sufficient.

Now Statement 2 states A is 3 units away from (0,0). Similarly it can said that A can be a circle with its center on (0,0) with a radius of 2. Equation: \(x^2 + y^2 = 4\). Again its insufficient alone.

As we combine them we know that two circles can intersect at two points of maybe just touch at one! A would be plausible only if the circles meet at one point. Now if the circles met at one point the distance between the centers = sum of the radii. Distance between centers = 5 units and clearly sum of radii is 5! Hence we can tell for sure that the circles touch at one point which is A. Therefore the answer is [C].

Hope it helps! :)

Regards,
A

Thanks! Just to confirm, the equation of the circles from statements 1 and 2 respectively would be :-

1) \((x-3)^2 + (y-4)^2 = 4\)
2) \(x^2 + y^2 = 9\)
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Re: What are the coordinates of point A?  [#permalink]

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New post 14 Jun 2013, 00:37
sps1604 wrote:
arpanpatnaik wrote:
sps1604 wrote:
What are the coordinates of point A?
1) A is 2 units away from (3,4)
2) A is 3 units away from (0,0)

How to solve this question?


The answer is [C]. The approach would be using the statements given.

We are to find the co-ordinates of point A. Hence they have to be a unique set of {x,y}.
Now Statement 1 states, A is 2 units away from (3,4). So basically all points at a distance of 2 units from (3,4) can be A. Hence the locus of Point A is a circle with the equation: \((x-3)^2 + (y-4)^2 = 9\). Clearly Statement 1 alone is not sufficient.

Now Statement 2 states A is 3 units away from (0,0). Similarly it can said that A can be a circle with its center on (0,0) with a radius of 2. Equation: \(x^2 + y^2 = 4\). Again its insufficient alone.

As we combine them we know that two circles can intersect at two points of maybe just touch at one! A would be plausible only if the circles meet at one point. Now if the circles met at one point the distance between the centers = sum of the radii. Distance between centers = 5 units and clearly sum of radii is 5! Hence we can tell for sure that the circles touch at one point which is A. Therefore the answer is [C].

Hope it helps! :)

Regards,
A

Thanks! Just to confirm, the equation of the circles from statements 1 and 2 respectively would be :-

1) \((x-3)^2 + (y-4)^2 = 4\)
2) \(x^2 + y^2 = 9\)


Yes.

In an xy coordinate system, the circle with center (a, b) and radius r is the set of all points (x, y) such that:
\((x-a)^2+(y-b)^2=r^2\)

Image

This equation of the circle follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram above, the radius is the hypotenuse of a right-angled triangle whose other sides are of length x-a and y-b.

If the circle is centered at the origin (0, 0), then the equation simplifies to: \(x^2+y^2=r^2\)

For more check: math-coordinate-geometry-87652.html

Hope it helps.
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Re: What are the coordinates of point A?  [#permalink]

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New post 14 Jun 2013, 01:28
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Quote:
Thanks! Just to confirm, the equation of the circles from statements 1 and 2 respectively would be :-
1) \((x-3)^2 + (y-4)^2 = 4\)
2) \(x^2 + y^2 = 9\)


That is correct! As Bunnel has already confirmed, the equation of the circle at center (p,q) is \((x-p)^2 + (y-q)^2 = r^2\)
I apologize I swapped the values, the equations are as you mentioned! :)

Regards,
A
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Re: What are the coordinates of point A?  [#permalink]

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New post 11 Oct 2016, 11:06
sps1604 wrote:
What are the coordinates of point A?

(1) A is 2 units away from (3,4)
(2) A is 3 units away from (0,0)


Question : What are the coordinates of point A?

Statement 1: A is 2 units away from (3, 4).
A can be anywhere on the circumference of a circle centered (3, 5) with Radius of 2 units, Hence
NOT SUFFICIENT


Statement 2: A is 3 units away from (0, 0)
A can be anywhere on the circumference of a circle centered (0, 0) with Radius of 3d units, Hence
NOT SUFFICIENT

Combining the two statements
There will be One point of Intersection of Circles drawn by the two statements (As shown in figure), hence
SUFFICIENT

Answer: Option C
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Re: What are the coordinates of point A?  [#permalink]

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Re: What are the coordinates of point A?   [#permalink] 23 May 2019, 04:06
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