apex231 wrote:In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?
(1) a/b = c/d
(2) sqrt((a)^2) + sqrt((b)^2) = sqrt((c)^2) + sqrt((d)^2)

Distance of (a, b) from origin = √(a² + b²)
Distance of (c, d) from origin = √(c² + d²)
They will be equidistant from the origin when these two quantities will be equal.

Statement 1: (a/b) = (c/d)
Implies, ad = bc
This is not enough to conclude whether (a, b) and (c, d) are equidistant from origin or not.

Not sufficient.

Statement 2: √(a²) + √(b²) = √(c²) + √(d²)
Implies, |a| + |b| = |c| + |d|
This is not enough to conclude whether (a, b) and (c, d) are equidistant from origin or not.

Not sufficient.

1 & 2 Together: Multiply the equation obtained from statement 2 by |d|.
=> |a||d| + |b||d| = |c||d| + |d||d|

Now from statement 1, ad = bc => |a||d| = |b||c|. Replace |b||c| instead of |a||d| in the above equation.
=> |b||c| + |b||d| = |c||d| + |d||d|
=> |b|*(|c| + |d|) = |d|*(|c| + |d|)
=> (|b| - |d|)(|c| + |d|) = 0

Note that for the ratio of statement 1 to be defined, b and d cannot be equal to zero. Hence (|c| + |d|) cannot be equal to zero. Which implies (|b| - |d|) = 0.

Therefore, |b| = |d|
This again implies |a| = |c| as |a||d| = |b||c|

This means the absolute values of the coordinates of the points are same. Which implies they are equidistant from the origin. This is because while determining the distance from origin we are squaring the coordinates and thus the sign doesn't matter.

Sufficient

The correct answer is C.