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# In a rectangular coordinate system, are the points (x, y) and (w, z)

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Joined: 02 Sep 2009
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In a rectangular coordinate system, are the points (x, y) and (w, z)  [#permalink]

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26 Sep 2016, 02:29
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Difficulty:

75% (hard)

Question Stats:

44% (01:34) correct 56% (01:50) wrong based on 72 sessions

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In a rectangular coordinate system, are the points (x, y) and (w, z) equidistant from the origin?

(1) |x|+|y|=|w|+|z|

(2) x/y=w/z

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Re: In a rectangular coordinate system, are the points (x, y) and (w, z)  [#permalink]

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26 Sep 2016, 03:41
Bunuel wrote:
In a rectangular coordinate system, are the points (x, y) and (w, z) equidistant from the origin?

(1) |x|+|y|=|w|+|z|

(2) x/y=w/z

points are equidistant from origin if they lie on the circumference of a circle with origin (0,0) or coincide.

equation of a circle passing through origin is x^2+ y^2 = r^2

question becomes is x^2+y^2 = w^2+z^2 or does they coincide

1 clearly insuff, try subst,

2
implies the 2 points are a translation of each other and are collinear or they coincide ...insuff

both together

the only way for the 2 conditions to occur is the points to coincide ... same distance from origin ...c
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Re: In a rectangular coordinate system, are the points (x, y) and (w, z)  [#permalink]

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02 Oct 2016, 10:04
Bunuel wrote:
In a rectangular coordinate system, are the points (x, y) and (w, z) equidistant from the origin?

(1) |x|+|y|=|w|+|z|

(2) x/y=w/z

(1) if x=y=w=z=2 then Yes
if x=2 y=2 && w=3 z=1 then No...........insuff........

(2) if x=y=w=z=2 then Yes
if x=4 y=2 && w=6 z=3.....No.....insuff

Combining from (2) coordinates (x,y) have same sign as (w,z)
from (1) x and y have same magnitude as w and z.....sufff

Ans C
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Re: In a rectangular coordinate system, are the points (x, y) and (w, z)  [#permalink]

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03 Oct 2016, 01:46
2
1
Bunuel wrote:
In a rectangular coordinate system, are the points (x, y) and (w, z) equidistant from the origin?

(1) |x|+|y|=|w|+|z|

(2) x/y=w/z

For two points to be equidistant from the origin, they must lie on the same circle with centre at origin.
So the question is this:
Is $$x^2 + y^2 = w^2 + z^2?$$
which is the same as "Is $$|x|^2 + |y|^2 = |w|^2 + |z|^2?$$"

(1) |x|+|y|=|w|+|z|
The sum of absolute values of the co-ordinates is the same. So the coordinates could be say, (1, 4) and (-1, 4) which lie on the same circle (because $$1^2 + 4^2 = (-1)^2 + 4^2$$) but they could also be (1, 4) and (2, 3) which do not lie on the same circle (because $$1^2 + 4^2$$ is not the same as $$2^2 + 3^2$$).

(2) x/y=w/z
x = yk and w = zk
The co-ordinates could be say, (1, 4) and (-1, -4) which lie on the same circle or they could be say, (1, 4) and (2, 8) which do not lie on the same circle.

Using both,
$$|yk| + |y| = |zk| + |z|$$
$$|y|*(|k| + 1) = |z|*(|k| + 1)$$
$$|y| = |z|$$
This means $$|x| = |w|$$ too.
Then $$|x|^2 + |y|^2 = |w|^2 + |z|^2$$
So (x, y) and (w, z) do lie on the same circle centered at (0, 0).

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Re: In a rectangular coordinate system, are the points (x, y) and (w, z)  [#permalink]

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30 Oct 2017, 14:05
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Re: In a rectangular coordinate system, are the points (x, y) and (w, z) &nbs [#permalink] 30 Oct 2017, 14:05
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