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

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In the rectangular coordinate system, are the points (a, b) [#permalink] New post 10 Apr 2010, 22:25
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In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

(1) \frac{a}{b}=\frac{c}{d}

(2) \sqrt{a^2}+ \sqrt{b^2} = \sqrt{c^2} + \sqrt{d^2}
[Reveal] Spoiler: OA

Last edited by Bunuel on 07 Apr 2012, 07:24, edited 1 time in total.
Edited the question and added the OA
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Re: ManhttanGMAT Practice CAT [#permalink] New post 11 Apr 2010, 00:11
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nsp007 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}

Will post OA later.



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.
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Re: ManhttanGMAT Practice CAT [#permalink] New post 08 Sep 2010, 01:22
Bunuel, had condition (2) simply said a+b=c+d (instead of the squares and square root) how would have the answer changed?

Also in the current question - is a^2+b^2=c^2+d^2?
When I square both sides of (2), I get a^2+b^2+2sqrt(a^2b^2)=c^2+d^2+2sqrt(c^2d^2)
so if ab=cd then this is satisfied. however (1) only gives me ad=bc, how do I infer ab=cd from that? I am following a different process, but I should end up with the same answer. Not sure where am I wrong?
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Re: ManhttanGMAT Practice CAT [#permalink] New post 09 Sep 2010, 00:42
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??
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Re: ManhttanGMAT Practice CAT [#permalink] New post 09 Sep 2010, 00:55
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amitjash wrote:
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).
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Re: points equidistant from origin? [#permalink] New post 16 Oct 2010, 14:52
(1) knowing these proportions does not help me solve it, because for example if 3/1 = 9/3 , point (a,b) will be closer to the origin than point (b,c)

(2) this statement tells that |a|+|b|=|c|+|d| , which is still not sufficient because we lack information about the correlation between |a| and |b|,and |c|and|d|.

But if we combine the two statements together we will have this correlation from statement (1) and then both statements taken together will be sufficient.

Answer should be C.
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Re: points equidistant from origin? [#permalink] New post 16 Oct 2010, 15:21
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Orange08 wrote:
In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

a. a/b = c/d

b. \sqrt{a^2}+\sqrt{b^2} = \sqrt{c^2} + \sqrt{d^2}


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)
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Re: Difficult Geometry DS Problem [#permalink] New post 12 Jan 2011, 18:43
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abmyers 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}

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.


You can take values to solve this question quickly:

Statement 1: a/b = c/d
(a,b) and (c,d) may be equidistant from the origin e.g. (1, 1) and (-1, -1) or they may not be e.g. (1, 1) and (2, 2). Not sufficient.

Statement 2: \sqrt{a^2}+\sqrt{b^2} = \sqrt{c^2} + \sqrt{d^2}
That is, |a|+|b|=|c|+|d|
(a,b) and (c,d) may be equidistant from the origin e.g. (1, 3) and (-1, -3) or they may not be e.g. (1, 3) and (2, 2)

Using both together, |a|+|b|=|c|+|d| and a/b = c/d.
This means that if a/c = 1/2, c/d cannot be 2/4 or -3/-6 etc. c/d has to be either 1/2 or (-1)/(-2). Similarly, if a/b = (-1)/2, c/d = (-1)/2 or 1/(-2))
Any pair of such points will be equidistant. Answer (C).
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Re: ManhttanGMAT Practice CAT [#permalink] New post 27 May 2012, 22:56
Bunuel wrote:
nsp007 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}

Will post OA later.



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.


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
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Re: ManhttanGMAT Practice CAT [#permalink] New post 04 Jun 2012, 18:39
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kunalbh19 wrote:

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.
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Re: In the rectangular coordinate system, are the points (a, b) [#permalink] New post 15 Jan 2013, 07:45
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.

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Re: ManhttanGMAT Practice CAT [#permalink] New post 22 Apr 2013, 11:55
Bunuel wrote:
nsp007 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}

Will post OA later.



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.

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?
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Re: ManhttanGMAT Practice CAT [#permalink] New post 22 Apr 2013, 21:00
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mun23 wrote:
\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
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Re: coordinate geometry [#permalink] New post 15 Aug 2013, 18:29
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sudharsansuski wrote:
In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

(1) a/b = c/d

(2) (a^2)^0.5 + (b^2)^0.5 = (c^2)^0.5 + (d^2)^0.5

I didnt understand the answer explanation given by MGMAT. Could someone please help.





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.
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Re: In the rectangular coordinate system, are the points (a, b) [#permalink] New post 16 Aug 2013, 02:33
sudharsansuski wrote:
In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

(1) a/b = c/d

(2) (a^2)^0.5 + (b^2)^0.5 = (c^2)^0.5 + (d^2)^0.5

I didnt understand the answer explanation given by MGMAT. Could someone please help.


basically we need to prove
a^2+b^2 = c^2+d^2

stmnt 1:
\frac{a}{b} = \frac{c}{d}

1/2 = 3/6 ==>using this we can prove NO
1/2 = 1/2 ==>using these numbers we can prove yes

hence insufficient
statement 2:
\sqrt{(a^2)} + \sqrt{(b^2)} = \sqrt{(c^2)} + \sqrt{(d^2)}or|a| + |b| = |c| + |d|

putting a=b=c=d=1..we will get YES.
Putting a= 1, b=2, c= 0, d=3...we will get NO.
HENCE INSUFFICIENT.

combining.
from 2nd statement: |a| + |b| = |c| + |d|
rearrange:
|a| - |d| = |c| - |b| and then square both sides.
a^2 + d^2 - 2*|a|*|d| = c^2 + b^2 - 2*|c|*|b|.

Since ad = bc as per statement 1
we can cancel few things:
a^2 + d^2 - 2*|a|*|d| = c^2 + b^2 - 2*|c|*|b|.

hence sufficient
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Re: In the rectangular coordinate system, are the points (a, b) [#permalink] New post 16 Aug 2013, 08:13
nsp007 wrote:
In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

(1) \frac{a}{b}=\frac{c}{d}

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


1. ad-bc=0 means that the vector pointing to (c,d) is a scalar multiple of the vector pointing to (a,b). In other words they are collinear with the origin. If they are equidistant to the origin, the scalar will be 1 or -1. But given this information it could be any scalar. Not sufficient.

2. |a|+|b|=|c|+|d|. This means that (a,b) and (c,d) both lay on some square centered at the origin. But they could be anywhere on the square. Not sufficient.

1 and 2. (a,b) and (c,d) are collinear with the origin and both lay on a square centered at the origin. Therefore they must be the same distance from the origin. Sufficient. C
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In the rectangular coordinate system, are the points (a, b) [#permalink] New post 14 Jul 2014, 04:52
In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

(1) a/b = c/d

(2) (a^2 + b^2)^(1/2) = (c^2 + d^2)^(1/2)


A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statements (1) and (2) TOGETHER are NOT sufficient.
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Re: In the rectangular coordinate system, are the points (a, b) [#permalink] New post 14 Jul 2014, 05:56
janxavier wrote:
In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

(1) a/b = c/d

(2) (a^2 + b^2)^(1/2) = (c^2 + d^2)^(1/2)


A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statements (1) and (2) TOGETHER are NOT sufficient.


Using distance formula, we can say that statement 2 is sufficient to answer that (a,b) and (c,d) are equidistant from Origin. Please check the OA posted.
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Re: In the rectangular coordinate system, are the points (a, b) [#permalink] New post 14 Jul 2014, 06:20
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janxavier wrote:
In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

(1) a/b = c/d

(2) (a^2 + b^2)^(1/2) = (c^2 + d^2)^(1/2)


A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statements (1) and (2) TOGETHER are NOT sufficient.


Merging similar topics. Please refer to the discussion above.
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Re: In the rectangular coordinate system, are the points (a, b) [#permalink] New post 31 Jul 2014, 04:33
This was my view:

You see immediately that both statements alone are not sufficient. So start evaluating the combination of both:

Statement A says that that the points are on the same line through the origin. Statement B says that the different "possible" points all have an equal distance to the x-axis, but also to the y-axis. Therefore, combining both options say that either (a,b) = (c,d) or (a,b) = (-c, -d). So equidistant from origin.
Re: In the rectangular coordinate system, are the points (a, b)   [#permalink] 31 Jul 2014, 04:33
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