Anybody - coordinate plane question; clarity please

[Vishnu88]

In a rectangular coordinate plane, are the points (a,b) and (c,d) equidistant from the origin?

1. a/b = c/d
2. √a^2 + √b^2 = √c^2 + √d^2


[spoiler]OA = C[/spoiler]

[cans]

A(a,b) , C(c,d), O(0,0)
is AO = CO??
a)a/b = c/d
insufficient. because according to this it can be possible that a=c and b=d in which case AO=CO
But it is also possible that a=2c and b=2d in which case AO>CO
b)AO^2 = (a-0)^2 + (b-0)^2 a^2 + b^2
CO^2 = c^2 + d^2
it means AO^2 = CO^2
AO = +-CO
but as both AO and CO are distances, both are positive and thus AO=CO
Sufficient
IMO B

[Frankenstein]

A(a,b) , C(c,d), O(0,0)
is AO = CO??
a)a/b = c/d
insufficient. because according to this it can be possible that a=c and b=d in which case AO=CO
But it is also possible that a=2c and b=2d in which case AO>CO
b)AO^2 = (a-0)^2 + (b-0)^2 a^2 + b^2
CO^2 = c^2 + d^2
it means AO^2 = CO^2
AO = +-CO
but as both AO and CO are distances, both are positive and thus AO=CO
Sufficient
IMO B

Hi,
Statement (2) : sqrt(a^2)+sqrt(b^2)+ = sqrt(c^2)+sqrt(d^2). You have mistakenly taken it as
a^2+b^2 = c^2+d^2.

[Frankenstein]

Hi,
Let A(a,b), B(c,d) and O(0,0)
From(1): Let a/b = c/d = m
So b/a = d/c.Slope of OA=slope of OB. So, O,A,B are on the same line.
Insufficient
From(2): |a|+|b| = |c|+|d|
if A(1,4) and B(2,3), A,B are not equidistant from origin
if A(1,2) and B(2,1), A,B are equidistant from origin
Insufficient
Both (1)&(2):|bm|+|b| = |dm|+|d| => |b|(1+|m|) = |d|(1=|m|)
So, |b| = |d| =>b^2 = d^2 and a^2 = c^2
So, a^2+b^2 = c^2+d^2 => OA^2 = OB^2
Sufficient

Hence, C

[cans]

A(a,b) , C(c,d), O(0,0)
is AO = CO??
a)a/b = c/d
insufficient. because according to this it can be possible that a=c and b=d in which case AO=CO
But it is also possible that a=2c and b=2d in which case AO>CO
b)AO^2 = (a-0)^2 + (b-0)^2 a^2 + b^2
CO^2 = c^2 + d^2
it means AO^2 = CO^2
AO = +-CO
but as both AO and CO are distances, both are positive and thus AO=CO
Sufficient
IMO B

Hi,
Statement (2) : sqrt(a^2)+sqrt(b^2)+ = sqrt(c^2)+sqrt(d^2). You have mistakenly taken it as
a^2+b^2 = c^2+d^2.

Thanks for correction
a)insufficient.
b) sqrt(a^2)+sqrt(b^2)+ = sqrt(c^2)+sqrt(d^2)
|a|+|b|=|c|+|d|
square both sides.
a^2 + b^2 + 2|ab| = c^2 + d^2 + 2|cd|
To find if OA=OC, we need to know if |ab|=|cd| (if yes, OA=OC and if not OA is not equal to OC)
we can't find whether |ab|=|cd|
Insufficient.
a*b together) a/b=c/d=m
a=bm;c=dm
Thus |bm| + |b| = |dm| + |d|
|b||m| + |b| = |m||d| + |d|
|b|=|d|
b=+-d
ab=bm*b = b^2*m = d^2*m = cd
thus |ab|=|cd|
Thus sufficient.
IMO C

[cans]

A(a,b) , C(c,d), O(0,0)
is AO = CO??
a)a/b = c/d
insufficient. because according to this it can be possible that a=c and b=d in which case AO=CO
But it is also possible that a=2c and b=2d in which case AO>CO
b)AO^2 = (a-0)^2 + (b-0)^2 a^2 + b^2
CO^2 = c^2 + d^2
it means AO^2 = CO^2
AO = +-CO
but as both AO and CO are distances, both are positive and thus AO=CO
Sufficient
IMO B

Hi,
Statement (2) : sqrt(a^2)+sqrt(b^2)+ = sqrt(c^2)+sqrt(d^2). You have mistakenly taken it as
a^2+b^2 = c^2+d^2.

Thanks for correction
a)insufficient.
b) sqrt(a^2)+sqrt(b^2)+ = sqrt(c^2)+sqrt(d^2)
|a|+|b|=|c|+|d|
square both sides.
a^2 + b^2 + 2|ab| = c^2 + d^2 + 2|cd|
To find if OA=OC, we need to know if |ab|=|cd| (if yes, OA=OC and if not OA is not equal to OC)
we can't find whether |ab|=|cd|
Insufficient.
a*b together) a/b=c/d=m
a=bm;c=dm
Thus |bm| + |b| = |dm| + |d|
|b||m| + |b| = |m||d| + |d|
|b|=|d|
b=+-d
ab=bm*b = b^2*m = d^2*m = cd
thus |ab|=|cd|
Thus sufficient.
IMO C

[vikram4689]

we want to know whether a2 + b2 = c2 + d2

a) a/b =c/d ;insufficient
b) squaring both sides
a2 + b2 + 2*sqrt(a2b2) = c2 + d2 + 2*sqrt(c2d2); insufficent

using both a=bk and c=dk substitute in above eqn. and you'll get b=d and therefore a=c

so a2 + b2 = c2 + d2

[[email protected]]

To find if OA=OC, we need to know if |ab|=|cd| (if yes, OA=OC and if not OA is not equal to OC)
we can't find whether |ab|=|cd|
Insufficient.
a*b together) a/b=c/d=m
a=bm;c=dm
Thus |bm| + |b| = |dm| + |d|
|b||m| + |b| = |m||d| + |d|
|b|=|d|
b=+-d
ab=bm*b = b^2*m = d^2*m = cd
thus |ab|=|cd|
Thus sufficient.


Cans GMAT Destroyer could you please explain me this part a bit more clearly...

[cans]

To find if OA=OC, we need to know if |ab|=|cd| (if yes, OA=OC and if not OA is not equal to OC)
we can't find whether |ab|=|cd|
Insufficient.
a*b together) a/b=c/d=m
a=bm;c=dm
Thus |bm| + |b| = |dm| + |d|
|b||m| + |b| = |m||d| + |d|
|b|=|d|
b=+-d
ab=bm*b = b^2*m = d^2*m = cd
thus |ab|=|cd|
Thus sufficient.


Cans GMAT Destroyer could you please explain me this part a bit more clearly...

sqrt(a^2)+sqrt(b^2)+ = sqrt(c^2)+sqrt(d^2)
|a|+|b|=|c|+|d|
square both sides.
a^2 + b^2 + 2|ab| = c^2 + d^2 + 2|cd| ---------eqn1
(OA=OC will mean OA^2=OC^2 or a^2 + b^2 = c^2 + d^2)
Thus to determine whether OA=OC, we need to know whether |ab|=|cd|. If yes, we can cancel them and thus OA=OC and if not, then it will mean that a^2 + b^2 is not equal to c^2 + d^2 and thus OA!=OC (OA is not equal to OC).
from |a| + |b| = |c| + |d| we can't determine whether |ab| = |cd|
(say case1 a=1,b=4,c=2,d=3 and case2: a=c=2 and b=d=3)
Thus insufficient.

Now if take both a) and b) together..
we know a/b=c/d. let this be equal to m.
Thus a/b=c/d=m
->a=bm;c=dm;
|a| + |b| = |c| + |d|
|bm| + |b| = |dm| + |d|
|b||m| + |b| = |d||m| + |d|
|b| = |d|
|a|=|bm|=|dm|=|c|
Thus |ab| = |cd|
Thus OA=OC
I hope its clear.