In the rectangular coordinate system, are the points (a,

Origin on coordinate system is (0,0). The question is if distance between (0,0) to (a,b) is same as distance between (0,0) to (c,d)

Case 1: - a/b = c/d

let a=7, b= 14 then a/b = 1/2.

So c/d fraction has to be 1/2. i.e. c can be 2 and d can be 4. or c can be 6 & d can be 12 or c can be 7 & d can be 14. So distance between origin can either be same to (c,d) or different from (a,b). Clearly insufficient

Case 2:- (a^2)^0.5 + (b^2)^0.5 = (c^2)^0.5 + (d^2)^0.5

translates to "a + b = c + d"

let (a,b) = (4,4) and (c,d) = (4,4). Then it satisfies the condition a+b = c+d. Also distance from origin to (a,b) is same as (c,d).

let (a,b) = (5,3) and (c,d) = (4,4). Then it satisfies the condition a+b = c+d. And distance from origin to (a,b) is not same as (c,d).

Insufficient.

Lets take 1 & 2 together

we have a/b = c/d.......so a= bc/d--- Eq (1)

we also have a + b = c + d

substitute a=bc/d from Eq(1)

bc/d + b = c + d

b(c/d + 1) = c + d

b(c + d) = d(c+d)

so b = d

similarly we get a=b.

so taking 1 & 2 together, the distance between origin to both points are equal.