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In the rectangular coordinate system, are the points (a, b) [#permalink]
 10 Apr 2010, 23:25
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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}
Last edited by Bunuel on 07 Apr 2012, 08:24, edited 1 time in total.
Edited the question and added the OA
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Re: ManhttanGMAT Practice CAT [#permalink]
11 Apr 2010, 01:11
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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]
11 Apr 2010, 01:20
Thanks Bunuel , for such a prompt response !!
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Re: ManhttanGMAT Practice CAT [#permalink]
08 Sep 2010, 02:22
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]
09 Sep 2010, 01:42
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]
09 Sep 2010, 01:55
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Re: points equidistant from origin? [#permalink]
16 Oct 2010, 15:52
(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.
What is the OG answer?
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Re: points equidistant from origin? [#permalink]
16 Oct 2010, 16:21
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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: points equidistant from origin? [#permalink]
17 Oct 2010, 02:06
shrouded1 wrote: (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.
[highlight](1+2) a/c=b/d=x say (needs, c,d to be non zero)[/highlight]
a = cx
b = dx
Answer is (c) Excellent. Didn't click me of equating a/c and b/d to x. I wonder in real GMAT test if such calculation intensive question do appear.
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Re: Difficult Geometry DS Problem [#permalink]
12 Jan 2011, 19:43
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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]
27 May 2012, 23:56
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]
04 Jun 2012, 19:39
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]
15 Jan 2013, 08:45
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]
22 Apr 2013, 12:55
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]
22 Apr 2013, 22:00
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: ManhttanGMAT Practice CAT
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