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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.

Re: points equidistant from origin? [#permalink]
16 Oct 2010, 15:21

3

This post received KUDOS

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

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: coordinate geometry [#permalink]
15 Aug 2013, 18:29

1

This post received KUDOS

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: ManhttanGMAT Practice CAT [#permalink]
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: points equidistant from origin? [#permalink]
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.

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)

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, 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.}

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?

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

gmatclubot

Re: In the rectangular coordinate system, are the points (a, b)
[#permalink]
16 Aug 2013, 08:13