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

(1) If a/b = c/d, a and b could be 100 and c and d could be 1 or a, b, c and d could be 1. NSF.

(2) If (a^2)^(1/2) + (b^2)^(1/2) = (c^2)^(1/2) + (d^2)^(1/2), then a+b = c+d. In this case, if a = 9 and b = 1, and c =d=5, a+b = c+d is true. but they have different distance.NSF.

From 1 and 2, a must be equal to c and b must be equal to d. So Suff.

Thats C.

in order for the points to be equidistant

a2 + b2 = c2+ d2 -----> eq X

1--> insufficent (has already been discussed)
2--> insufficent (has already been discussed)

combining both,
equation 2 can be written as a+b = c+d

eq1 a/b = c/d
a+b/b = c+d/d, from eq 1 from b=d ----3

similarly, b/a = d/c, a+b/a = d+c/c from eq 1 a = c ----4


plug in values of a, b in eq X... it satifies.

Hence points are equidistant. Answer is C

I have a question here bunuel. How did you get a = c * x and b= d*x??

Because from this it means that x = b/d = a/c

Can you please explain??

It's the same: \(\frac{b}{d} = \frac{a}{c}\) --> \(bc=ad\)--> \(\frac{c}{d}=\frac{a}{b}\).

Given: \(\frac{a}{b}=\frac{c}{d}=\frac{cx}{dx}\) (as the ratios are equal then there exist some \(x\) for which \(a=cx\) and \(b=dx\)).

Distance of \((x,y)\) from origin is \(\sqrt{x^2+y^2}\)
So we need to answer \(\sqrt{a^2+b^2}=\sqrt{c^2+d^2}\) ?

(1) a/b=c/d ... doesnt really help in proving or disproving. Insufficient

(2) This is equivalent to saying \(|a|+|b|=|c|+|d|\). Again insufficient to say anything about the statement we have.

(1+2) a/c=b/d=x say (needs, c,d to be non zero)

a = cx
b = dx

|a|+|b|=|c|+|d|
|cx| - |c| = |d| - |dx|
|c|(|x|-1)=|d|(1-|x|)
(|c|+|d|)(|x|-1)=0
Since c,d are non-zero means |x|=1
So either a=c & b=d OR a=-c or b=-d
Either case a^2=c^2 and b^2=d^2
Hence \(\sqrt{a^2+b^2}=\sqrt{c^2+d^2}\)
Sufficient

Answer is (c)

Official explanation says:
"If we know the proportion of a to b is the same as c to d and that |a| + |b| = |c| + |d|, then it must be the case that |a| = |c| and |b| = |d| ?

could someone elaborate how we are supposed to know |a| = |c| and |b| = |d|? a bit vague statement for me

thanks for responses

a/b=c/d <---- Equation 1

|a|+|b|=|c|+|d|

Add 1 to both sides of Equation 1

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

Now a+b=c+d => 1/b=1/d

Thus, b=d
Similarly,
b/a=d/c
Adding 1 to both sides of above equation:
(b+a)/a=(d+c)/c
a+b=d+c => 1/a=1/c
Therefore a=c

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

Distance between the point A (x,y) and the origin can be found by the formula: \(D=\sqrt{x^2+y^2}\).

So we are asked whether \(\sqrt{a^2+b^2}=\sqrt{c^2+d^2}\)? Or whether \(a^2+b^2=c^2+d^2\)?

(1) \(\frac{a}{b}=\frac{c}{d}\) --> \(a=cx\) and \(b=dx\), for some non-zero \(x\). Not sufficient.

(2) \(\sqrt{a^2}+\sqrt{b^2}=\sqrt{c^2} +\sqrt{d^2}\) --> \(|a|+|b|=|c|+|d|\). Not sufficient.

(1)+(2) From (1) \(a=cx\) and \(b=dx\), substitute this in (2): \(|cx|+|dx|=|c|+|d|\) --> \(|x|(|c|+|d|)=|c|+|d|\) --> \(|x|=1\) (another solution \(|c|+|d|=0\) is not possible as \(d\) in (1) given in denominator and can not be zero, so \(d\neq{0}\) --> \(|c|+|d|>0\)) --> now, as \(|x|=1\) and \(a=cx\) and \(b=dx\), then \(|a|=|c|\) and \(|b|=|d|\) --> square this equations: \(a^2=c^2\) and \(b^2=d^2\) --> add them: \(a^2+b^2=c^2+d^2\). Sufficient.

Answer: C.

your explanation is great as always ! thaaanks
+ Kudo

No that's not correct.

|a|+|b|=|c|+|d| doesn't mean that a+b=c+d (consider a=b=1 and c=d=-1). So from |a|+|b|=|c|+|d| and a/b=c/d we can not derive that a=c and b=d. What we can derive is that |a|=|c| and |b|=|d|. Refer to my post above for complete solution.

Here's a simpler solution......

We need to prove a^2 + d^2 = c^2 + b^2

Simpler way is to rearrange statement 2 viz. |a| + |b| = |c| + |d| as
1. |a| - |d| = |c| - |b| and then square both sides. We get -
2. a^2 + d^2 - 2*|a|*|d| = c^2 + b^2 - 2*|c|*|b|.

Since ad = bc as per statement 1,

3. |ad| = |bc| => |a|.|d| = |b|.|c| (Rule -> Abs of Product = Product of Abs)

So we can cancel the third term out from both sides of equation 2. to get the desired equation

HTH

I am getting a different final result for answer (C). Here is my approach:
\(\sqrt{a^2} - \sqrt{d^2} = \sqrt{c^2} - \sqrt{b^2}\) ----from statement (2)

Squaring both sides

\(a^2 + d^2 - 2\sqrt{a^2*d^2} = c^2 + b^2 - 2\sqrt{b^2*c^2}\)

from statement (1) we know that ad=bc --> a^2*d^2 = b^2*c^2

Cancelling last term of both sides, we get

a^2 + d^2 = c^2 + b^2
Thus, a^2 - b^2 = c^2 - d^2

Not able to figure out where I went wrong. Please suggest!

Thanks

There is nothing wrong with your approach. The relation you have got holds (a^2 - b^2 = c^2 - d^2).
But if you start doing algebraic manipulations on the starting point without an eye on what you need to achieve at the end, you may not obtain the result you want. You can manipulate an expression in many ways to get seemingly different results.

Notice that Bunuel got a^2 = c^2 and b^2 = d^2. He could have chosen to add them as
a^2 + d^2 = c^2 + b^2 (or subtract them) which is same as your result. But he chose to add them as a^2 + b^2 = c^2 + d^2 to get the expression he desired. Using the same two expressions, Bunuel arrived at a^2 + b^2 = c^2 + d^2 and you arrived at a^2 - b^2 = c^2 - d^2, both of which are correct. But only one of them (a^2 + b^2 = c^2 + d^2) helps you answer the question directly since you know that the distance of a point (a, b) is given by square root of (a^2 + b^2).

Try to find the relation between individual variables instead of working on the equations as a whole. Either use number plugging or Bunuel's algebraic approach.

Absolutely, I found this approach much better. Infact I am wondering why we are targetting absolute values in the equation.

(a+b)^2 = a^2+b^2+2ab [no absolute |a|, |b| needed]

From
1) we know ab = cd
2) we know a + b = c + d. and hence (a+b)^2 = (c+d)^2

Combining 1) & 2) we can very well see that a^2+b^2 = c^2+d^2

1+2:

1) Ensures that the lines joining each of the two points to origin have same slope.
2) Ensures that absicca and ordinates correspondingly have equal magnitudes.
{If one line has points (a,b) then the other line will have coordinates (ak,bk) but (2) ensures that |k| = 1.}

Hence C.

KUDOS IF YOU LIKE!

Hi Bunnel
i am not understanding how \(\sqrt{a^2}+\sqrt{b^2}=\sqrt{c^2} +\sqrt{d^2}\) --> \(|a|+|b|=|c|+|d|\)
have come?

By convention, only positive roots are considered (at least in GMAT)

\(\sqrt{4} = 2\) (and not -2)

Similarly, \(\sqrt{a^2} = |a|\) i.e. only the positive value