If x2+5y2+z2=2y(2x+z),
then which of the following statements are necessarily true?
I. x=2y
II. x=2z III. 2x=z
(1)only I
(2)only II (3)only III (4)both I and II
Solution follows here:
Solution:
This is an algebraic manipulation and simplification
problem;
x2+5y2+z2=2y(2x+z)
=> x2+5y2+z2=4xy+2yz
x2+5y2+z2-4xy-2yz=0
=> x2+4y2+y2+z2-4xy-2yz=0
observe 4xy can be written as 2(x)(2y) and 2yz
can be written as 2(y)(z)
x2+(2y)2-2(x)(2y)-2(x)(y)+y2+z2-2yz
=0
(x-2y)2+(y-z)2 = 0
This specifies sum of squares of two terms is
zero
=> both the terms are inevitably zero (it is
not possible if the terms are +ve or –ve,
both the terms must be zero)
=> x-2y = 0; y-z = 0
=> x=2y; y=z
Only x=2y is given in the options.
Answer (1)
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