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
Explanation:
x² + 5y² + z² = 4xy + 2yz
∴ x² + 4y² + y² + z² – 4xy – 2yz = 0
∴ (x² + 4y² – 4xy) + (y² – 2yz + z²) = 0
∴ (x – 2y)² + (y – z)² = 0 …(i)
As per equation (i),
∴ (x – 2y)² = 0 and (y – z)² = 0
∴ x = 2y and y = z
Hence, option (d).
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