Question 14 - End-of-Chapter Exercises - Chapter 3 Class 9 - The World of Numbers (Ganita Manjari I)

Transcript

Question 14 Three rational numbers x, y, z satisfy x + y + z = 0 and xy + yz + zx = 0. Show that all the rational numbers x, y, z must be simultaneously zero. We have an identity with 3 variables, it is (𝒙+𝒚+𝒛)^𝟐=𝒙^𝟐+𝒚^𝟐+𝒛^𝟐+𝟐(𝒙𝒚+𝒚𝒛+𝒛𝒙) Given that x + y + z = 0 & xy + yz + zx = 0 Putting these two values in our identity (𝟎)^𝟐=𝒙^𝟐+𝒚^𝟐+𝒛^𝟐+𝟐(𝟎) ■(@0=𝑥^2+𝑦^2+𝑧^2 ) ■(@𝒙^𝟐+𝒚^𝟐+𝒛^𝟐=𝟎) Now, we know that Square of any number ≥𝟎 So, 𝒙^𝟐≥𝟎,𝒚^𝟐≥𝟎, and 𝒛^𝟐≥𝟎 But, if the sum of squares is 0 It means the squares of each number is 0 Thus, 𝒙^𝟐=𝟎, 𝒚^𝟐=𝟎, and 𝒛^𝟐=𝟎 Since square of each number is 0, individually the numbers would be 0 Therefore, 𝒙=𝟎, 𝒚=𝟎, and 𝒛=𝟎 Thus, 𝒙,𝒚, and 𝒛 must all be simultaneously zero Hence proved