I think answer should be E
i arrived using the distance formula
experts needed here
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bblast
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anshumishra
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Question : a^2 + b^2 =c^2 + d^2 ?ankurmit 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
OA LAter
Statement 1:
a/b = c/d
If (a=2,b=3) and (c=-2,d=-3), then a^2 + b^2 = c^2 + d^2.
If (a=2,b=3) and (c=4, d=6), then a^2 +b^2 ≠c^2 + d^2.
Insufficient
Statement 2:
squaring both sides => a^2+b^2+2ab = c^2+d^2+2cd
So, if ab = cd, then the equality will hold. --- Not Sufficient
Combining 1 and 2 :
|a|+|b| = |c|+|d|
=>|a||d|+|b||d| = |c||d|+|d||d|
Fom 1, ad=bc, so |a||d| = |b||c|
=>|b||c|+|b||d| = |c||d|+|d||d|
=>|b| (|c|+|d|) = |d| (|c|+|d|)
Hence, |b| = |d|, and using 1 with this, |a| = |c|
Hence, C, since the points are equidistant from (0,0) or Origin.
Last edited by anshumishra on Sun Jan 16, 2011 6:08 am, edited 3 times in total.
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Rahul@gurome
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Distance of (a, b) from origin = √(a² + b²)ankurmit 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
Distance of (c, d) from origin = √(c² + d²)
They will be equidistant from the origin when these two quantities will be equal.
Statement 1: (a/b) = (c/d)
Implies, ad = bc
This is not enough to conclude whether (a, b) and (c, d) are equidistant from origin or not.
Not sufficient.
Statement 2: √(a²) + √(b²) = √(c²) + √(d²)
Implies, |a| + |b| = |c| + |d|
This is not enough to conclude whether (a, b) and (c, d) are equidistant from origin or not.
Not sufficient.
1 & 2 Together: Multiply the equation obtained from statement 2 by |d|.
=> |a||d| + |b||d| = |c||d| + |d||d|
Now from statement 1, ad = bc => |a||d| = |b||c|. Replace |b||c| instead of |a||d| in the above equation.
=> |b||c| + |b||d| = |c||d| + |d||d|
=> |b|*(|c| + |d|) = |d|*(|c| + |d|)
=> (|b| - |d|)(|c| + |d|) = 0
Note that for the ratio of statement 1 to be defined, b and d cannot be equal to zero. Hence (|c| + |d|) cannot be equal to zero. Which implies (|b| - |d|) = 0.
Therefore, |b| = |d|
This again implies |a| = |c| as |a||d| = |b||c|
This means the absolute values of the coordinates of the points are same. Which implies they are equidistant from the origin. This is because while determining the distance from origin we are squaring the coordinates and thus the sign doesn't matter.
Sufficient
The correct answer is C.
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Quant Expert
Gurome, Inc.
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On MBA sabbatical (at ISB) for 2011-12 - will stay active as time permits
1-800-566-4043 (USA)
+91-99201 32411 (India)
