Algebra-12 (CAT-2002)

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)