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).

Answer (C)

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Karishma

Veritas Prep GMAT Instructor

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