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taken to be as the GM

by sanju09 » Sat Sep 04, 2010 3:02 am
√(x y) is taken to be as the GM of x and y. If a b + b c is the GM of a^2 + b^2 and b^2 + c^2, then which of the following is correct?
(A) a + c = 2 b
(B) 1/a + 1/c = 2/b
(C) a c = b^2
(D) a + b + c = 0
(E) a + c = b/2


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by this_time_i_will » Sat Sep 04, 2010 4:11 am
IMO C
SQrt((a2+b2)(b2+c2)) = ab+bc
Squaring both sides...
(a2+b2)(b2+c2) = b2(a+c)^2
a2b2+a2c2+b2b2+b2c2 = b2a2+b2c2+2b2ac
a2c2+b2b2= 2b2ac
a2c2+b2b2-2b2ac = 0
(ac-b2)^2 = 0
(ac-b^2) = 0
b^2 = ac.

note: a2 = a^2 etc.
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by sanju09 » Sat Sep 04, 2010 4:28 am
this_time_i_will wrote:IMO C
SQrt((a2+b2)(b2+c2)) = ab+bc
Squaring both sides...
(a2+b2)(b2+c2) = b2(a+c)^2
a2b2+a2c2+b2b2+b2c2 = b2a2+b2c2+2b2ac
a2c2+b2b2= 2b2ac
a2c2+b2b2-2b2ac = 0
(ac-b2)^2 = 0
(ac-b^2) = 0
b^2 = ac.

note: a2 = a^2 etc.
nice one
The mind is everything. What you think you become. -Lord Buddha



Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
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