I'll change the letters in the question, because in coordinate geometry, x and y are usually reserved for variables (as in the equation of a line). So if we have the question:

Are (a, b) and (c, d) equidistant from the origin?
1. |a| + |b| = |c| + |d|
2. a/b = c/d

Neither statement is sufficient alone; certainly the points can be equidistant from the origin, because they could be the same point. But for Statement 1 we could have the points (7, 0) and (3, 4), say, and by 3-4-5 triangles, (3, 4) is 5 units from the origin, while (7, 0) is 7 units from the origin. For Statement 2, one point could be (1, 1) and the other (1000, 1000), which are very different distances from (0, 0).

Using only Statement 2, conceptually it's easier to take reciprocals: b/a = d/c. Now b/a is equal to some number m. But if b/a = m, then b = ma, and the point (a, b) is on the line y = mx. Since b/a = d/c = m, we also have d = mc, and the point (c, d) is also on the line y = mx. So from Statement 2 alone, our two points are on the same line passing through the origin. And conceptually, if you start at the origin and travel along a line y = mx, the larger |x| gets, the further away from the origin you get. So if |a| + |b| = |c| + |d| here, then |a| = |c| must be true, and the points must be equidistant from (0, 0) if we use both Statements (they're either the same point, or one is the reflection through the origin of the other). Or you could do algebra: we know b = ma and d = mc, so substituting into Statement 1, we find

|a| + |b| = |c| + |d|
|a| + |ma| = |c| + |mc|
|a| + |m|*|a| = |c| + |m|*|c|
|a|(1 + |m|) = |c|(1 + |m|)
|a| = |c|

I've ignored some technicalities about zero, but if one of the x-coordinates is zero, the two Statements are clearly sufficient (and the question needs to rule out the possibility that the y-coordinates are zero since we divide by them in Statement 2).
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statement (1) |x|+|y|=|w|+|z|

Take (2,2) and (1,3)
and (2,2) and (-2,-2) first set is not equidistant; However, the second is
Not sufficient

Statement (2) x/y=w/z
Take (2,2) and (1,1)
and (2,2) and (-2,-2) first set is not equidistant; However, the second is
Not sufficient

Combined

Set (2,4)(-2,-4)
(-2,4)(2,-4)

Such set of points will satisfy both statements combined i.e points which are mirror images over X, Y axis or Origin and such points will always be equidistant

Sufficient
(C) Is the answer IMO

IMO E

We need to prove \(x^2+y^2 = w^2+z^2\)

Let us consider statement 1 :

|x|+|y|=|w|+|z| => |x|-|z| = |w|-|y|
=> \((|x|-|z|)^2 = (|w|-|y|)^2\)
=> \(|x|^2 + |z|^2 - 2|x||z| = |w|^2 + |y|^2 -2|w||y|\)

So we cannot derive anything from this statement alone.

Now let's use statement 2:
\(x/y = w/z\) => xz = wy

Using both these statements we can say \(|x|^2 + |z|^2=|w|^2 + |y|^2\)

\(|x|^2 - |y|^2 =|w|^2 - |z|^2\), Hence we cannot prove anything even by using both the statements. The only case possible to do so is |x| = |z| & |y| = |w|.

In a rectangular coordinate system, are the points (x, y) and (w, z) equidistant from the origin?
So, question is asking, is \(x^2 +y^2 = w^2 + z^2 \)?

Stat1: |x|+|y|=|w|+|z|
or, |x|-|z|=|w|-|y| or, \(x^2 +z^2\) -2zx= \(w^2 +y^2\) -2yw. Not sufficient

Stat2: x/y=w/z, so, slope of (x, y) and (0,0) equals to slope of (w, z) and (0,0). But, we don't have an idea for distance. Not sufficient

Combining both, both points (x, y) and (w, z) have same slope and sum of absolute value of co-ordinates are equal too. Which makes points same or mirror image. So, distance from origin will be equal.

So, I think C.