The determinant $\left|\matrix{xp+y & x & y \cr yp+z & y & z \cr 0 & xp+y & yp+…

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The determinant $\left|\matrix{xp+y & x & y \cr yp+z & y & z \cr 0 & xp+y & yp+z}\right| = 0$ if

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Expand determinant and simplify ⇒ condition reduces to $y^2 = xz$ This is condition of GP Final Answer: $\boxed{x,y,z \text{ are in GP}}$

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