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In the rectangular coordinate system, are the points (a, 15 Aug 2013, 19:29
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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.
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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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In the rectangular coordinate system, are the points (a, 11 Aug 2014, 00:37
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Bunuel
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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

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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Bunuel, how did you quickly determine that (2) was Not Sufficient?
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In the rectangular coordinate system, are the points (a, 12 Aug 2014, 08:09
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Bunuel
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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

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.

Bunuel, how did you quickly determine that (2) was Not Sufficient?
Show more

You can do this with number plugging. The question asks whether \(a^2+b^2=c^2+d^2\) and (2) says that \(|a|+|b|=|c|+|d|\). If a = b = c = d = 0, then the answer would be YES but if a = 0, b = 2, c = 1 and d = 1, then the answer would be NO.
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In the rectangular coordinate system, are the points (a, 06 Jan 2015, 23:12
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Bunuel Sir,

I Didn't understand this:-

b/a=a/c->bc=ad->c/a=a/b
given a/b=c/d=cx/dx

(as the ratios are equal then there exist some for which d=cx and b=dx)
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In the rectangular coordinate system, are the points (a, 07 Jan 2015, 07:07
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manojpandey80
Bunuel Sir,

I Didn't understand this:-

b/a=a/c->bc=ad->c/a=a/b
given a/b=c/d=cx/dx

(as the ratios are equal then there exist some for which d=cx and b=dx)
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a/b = c/d = cx/dx

For example: 16/32 = 2/4 = (2*8)/(4*8).
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In the rectangular coordinate system, are the points (a, 24 Sep 2019, 10:43
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Kinshook
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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) \(\sqrt{a^2} + \sqrt{b^2} = \sqrt{c^2} + \sqrt{d^2}\)
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Asked: In the rectangular coordinate system, are the points \((a, b)\) and \((c, d)\) equidistant from the origin?
Q. a^2 + b^2 = c^2 + d^2 ?

(1) \(\frac{a}{b} =\frac{c}{d}\)
a = cx; b = dx
NOT SUFFICIENT

(2) \(\sqrt{a^2} + \sqrt{b^2} = \sqrt{c^2} + \sqrt{d^2}\)
|a| + |b| = |c| + |d|
Squaring both sides
a^2 + b^2 + 2|a||b| = c^2 + d^2 + 2 |c||d|
NOT SUFFICIENT

(1) + (2)
(1) \(\frac{a}{b} =\frac{c}{d}\)
a = cx; b = dx
|a| = |x||c|; |b| = |x||d|
(2) \(\sqrt{a^2} + \sqrt{b^2} = \sqrt{c^2} + \sqrt{d^2}\)
|a| + |b| = |c| + |d|
|x||c| + |x||d| = |c| + |d|
|x| (|c| + |d|) = |c| + |d|
|x| = 1
|a| = |c|
&
|b| = |d|
a^2 + b^2 + = c^2 + d^2
SUFFICIENT

IMO C
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In the rectangular coordinate system, are the points (a, 23 Aug 2020, 23:13
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@Bunnel VeritasKarishma

I did this way. Could you pl. point out the mistake?

(1) + (2)

Given from St. 2: |a| + |b| = |c| + |d|

Squaring |a| + |b| = |c| + |d|, we get a^2 + b^2 + 2|ab| = c^2 + d^2 + 2|cd| ---(1)

Assuming that the question is correct, then a^2 + b^2 = c^2 + d^2

Thus, from eqn (1), we must have 2|ab| = 2|cd| => |ab| = |cd| ---(2)

From St. 1 a/b = c/d, we have a = bk and c = dk; where k is a positive int.

From eqn (2), we have b^2k = d^2k => |b| = |d|.

Now to conclude sufficient from this point?
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In the rectangular coordinate system, are the points (a, 27 Aug 2020, 00:10
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@Bunnel VeritasKarishma

I did this way. Could you pl. point out the mistake?

(1) + (2)

Given from St. 2: |a| + |b| = |c| + |d|

Squaring |a| + |b| = |c| + |d|, we get a^2 + b^2 + 2|ab| = c^2 + d^2 + 2|cd| ---(1)

Assuming that the question is correct, then a^2 + b^2 = c^2 + d^2

Thus, from eqn (1), we must have 2|ab| = 2|cd| => |ab| = |cd| ---(2)

From St. 1 a/b = c/d, we have a = bk and c = dk; where k is a positive int.

From eqn (2), we have b^2k = d^2k => |b| = |d|.

Now to conclude sufficient from this point?
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|b| = |d|

Put in |a| + |b| = |c| + |d| (from Statement 2) to get:
|a| + |b| = |c| + |b|
|a| = |c|

So we have b^2 = d^2 and a^2 = c^2 (by squaring the equations obtained)
Add to get a^2 + b^2 = c^2 + d^2
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In the rectangular coordinate system, are the points (a, 27 Aug 2020, 03:17
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mbaMission
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) \(\sqrt{a^2} + \sqrt{b^2} = \sqrt{c^2} + \sqrt{d^2}\)
Show more

Stmt-01-case-01
(a,b)=(c,d)= (2,2); ratios are same and they are equidistant.
Stmt-01-case-02
(a,b)=(50,100);(c,d)=(1,2);
Ratios are same but they are not equidistant.

Stmt-02
This can be simplified as |a|+|b|=|c|+|d|
This is will work when,
(a,b)=(c,d)= (2,3)
(a,b)=(4,1)&(c,d)= (2,3)
Hence, not sufficient.

Stmt 1&2
|a|+|b|=|c|+|d|
And
|a|*|d|=|c|*|d|
We can determine that they are equidistant or they are the same points.

Posted from my mobile device
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In the rectangular coordinate system, are the points (a, 12 Sep 2020, 23:25
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Bunuel
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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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Hello Bunuel
The explanation looks great! However, I did not understand this last step-
now, as \(|x|=1\) and \(a=cx\) and \(b=dx\), then \(|a|=|c|\) and \(|b|=|d|\)
Could you please explain this in detail or with concept that has been used here?
Also, could you please tell me the difference between |a|=|c| and a = |c| or |a|=c?
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In the rectangular coordinate system, are the points (a, 16 Jun 2023, 03:54
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My take-

From first option, a/b=c/d=x
a=bx, c=dx
Now for two points (a,b) & (c,d) to be equidistant from origin the distance of both these points from origin (0,0) must be equal.

Distance of (a,b) from origin is sqrt(a^2+b^2)=sqrt(b^2*x^2+b^2)= b*sqrt (x^2+1)
Similarly for point (c,d) distance is d*sqrt (x^2+1).
Since we don't know whether b or d are equal from option (1) hence insufficient.

From option (2), we can derive that
a+b=c+d (I have considered them as positive for convenience), other wise it would be |a|+|b|=|c|+|d|

so putting values it would come as, b(x+1)=d(x+1), it will come as |b|=|d|
again we don't the values of b & d hence this also is insufficient.

Now combining above, putting b=d in a+b=c+d will give a=c or |a|=|c| hence the points are equidistant from origin.
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In the rectangular coordinate system, are the points (a, 16 Jun 2023, 11:28
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there is a quicker what to solve this:
1st statement says that a/b = c/d. what does this mean geometrically? it means that the line "r" that passes through point (a,b) and origin, and the line "s" that passes through point (c,d) and origin, have the same inclination since the tangent is the same. In other words, they lay on the same line. Is that enough to say these points are equidistant from the origin? No, because we can have one point as close as we want from the origin, and the other as distant as we want.
2nd statement says that these points belong to a family of squares in which the vertices are in the axis, and the center in the origin (just plot |x|+|y|=10 and you will see it). Is that enough to say the points are equidistant? No, because a point in the vertice is more distant than a point in the line x=y or x=-y.
But if you combine both statements, given the symmetry, it is immediate they are equidistant from the origin.
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In the rectangular coordinate system, are the points (a, 15 Nov 2025, 13:08
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