Akash,

The answer is C

Statement 1 tells me that the proportion is equal. It does not mean that the points are equidistant.

For example if a=4 and b = 2 and c =2 and d = 1, a/b=c/d. But they are not equidistant.
For example if a=2 and b = 1 and c =2 and d = 1, a/b=c/d. But they are equidistant.

Insufficient.

Statement 2 tells me that mod a + mod b = mod c + mod d

This is insufficient

For example if mod a=10 and mod b = 2 and mod c =6 and mod d = 6, mod a + mod b = mod c + mod d. But they are not equidistant.
For example if mod a=10 and mod b = 2 and mod c =6 and mod d = 6, mod a + mod b = mod c + mod d. But they are equidistant.


Combined, the proportions must be equal and they must be equal in value as in statement 2. Sufficient.

In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?
(1) a/b = c/d
(2) √(a²) + √(b²) = √(c²) + √(d²)

Statement 1: a/b = c/d
If the equation is 1/2 = 1/2, then (1,2) and (1,2) are equidistant from the origin.
If the equation is 1/2 = 2/4, then (1,2) and (2,4) are not equidistant from the origin.
INSUFFICIENT.

Statement 2: √(a²) + √(b²) = √(c²) + √(d²)
√x² = |x|.
Rephrasing the statement, we get:
|a| + |b| = |c| + |d|.
If the equation is |0| + |2| = |0| + |2|, then (0,2) and (0,2) are equidistant from the origin.
If the equation is |0| + |2| = |1| + |1|, then (0,2) and (1,1) are not equidistant from the origin.
INSUFFICIENT.

Statements 1 and 2 combined:
Let a/b = c/d = 2.
Then a=2b and c=2d.
Substituting a=2b and c=2d into |a|+|b| = |c|+|d|, we get:
|2b|+|b| = |2d|+|d|
3|b| = 3|d|
|b| = |d|, implying that |a| = |c|.
Since the x values in (a,b) and (c,d) are equidistant from the origin, and the y values in (a,b) and (c,d) are equidistant from the origin, the two points are equidistant from the origin.
By extension, if a/b = c/d = k, where k is any nonzero value, the result will be the same.
SUFFICIENT.

The correct answer is C.