Guarantee Toughie(700+)-Creative Ways?-cordinate plane

[kaps786]

Toughie from MGMAT - looking for creative and different approaches to approach this problem.

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)


OA follows..





OA is C.

I have seen a couple of nice approaches on number plugging, but just checking if there are any other innovative ideas to think conceptually on this. I tried algebra...then tried to switch mid-way to plugging and messed it on time...

Pointers on CONCEPT, GUESSING, TAKEAWAYS on how to recognize what approach to adopt,,... all welcome.


Thank you

[gmatboost]

Plugging in is definitely best (and pretty quick) for the statements on their own.
If you choose the same value for each variable, they will be equidistant. To satisfy each statement but have them be non-equidistant, you can choose something like
1. 4, 4, 2, 2
2. 4, 1, 2, 3

When you combine them, if you're willing to make the uncomfortable assumption that a, b, c, and d are all positive, you can proceed with algebra. I say uncomfortable because it's not exactly true, but the sign doesn't affect the distance from the origin, so it won't matter.

a + b = c + d
d = a + b -c
ad = bc
a(a + b - c) = bc
a^2 + ab - ac = bc
a^2 + ab = bc + ac
a(a+b) = c(a+b)
a = c
ad = bc = ba
ad = ba
d = b
Since a=c, b=d, equidistant

Probably better not to go this way though.

[imhimanshu]

Hi GMAT Boost,

This is how I approached this question-

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)

Problem Statement -
If (a, b) and (c, d) are equidistant (o,o)

Then, a2+b2 = c2+d2 (a2 as "a square")

statement 1 -> ad=bc Insufficient

statement 2 - a+b =c+d
squaring both sides
a2+b2+2ab = c2+d2+2cd
so, if we could find if 2ab = 2cd or to be specific ab = cd, we could say whether pts are equidistant or not.

but as per 1,ad=bc. This is insufficient as per me. Could you please correct me where I am wrong.

Thanks

[Anurag@Gurome]

statement 2 - a+b =c+d
squaring both sides
a2+b2+2ab = c2+d2+2cd
so, if we could find if 2ab = 2cd or to be specific ab = cd, we could say whether pts are equidistant or not.

but as per 1,ad=bc. This is insufficient as per me. Could you please correct me where I am wrong.

Careful!
Statement 2 implies |a| + |b| = |c| + |d|
Also you never found that ab ≠ cd so you cannot say they are insufficient together.

If you're looking for a detailed mathematical solution, refer to this post by Rahul : https://www.beatthegmat.com/mgmat-cat-t73318.html#331635

[kaps786]

Hi Anurag

Thanks... Just pasting the solution by Rahul you referred to here.

I didnt get one step that was done in solution below, you maybe able to shed some light.

" Now from statement 1, ad = bc => |a||d| = |b||c|. "...How does this happen.. and why do we need to write it as |a||d=|b||c|




Solution



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