If three distinct points $P (3 u^{2}, 2 u^{3}); Q (3 v^{2}, 2 v^{3}); R (3 w^{2}, 2 w^{3})$ are collinear, then $u v + v w + w u$ is equal to ........

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Solution

Three distinct points $P (3 u^{2}, 2 u^{3}); Q (3 v^{2}, 2 v^{3}); R (3 w^{2}, 2 w^{3})$ are collinear.
$\Rightarrow ∣ ∣ ∣ ∣ ∣ \begin{matrix} 3 u^{2} & 2 u^{3} & 1 3 v^{2} & 2 v^{3} & 1 3 w^{2} & 2 w^{3} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$
$\Rightarrow ∣ ∣ ∣ ∣ ∣ \begin{matrix} u^{2} & u^{3} & 1 v^{2} & v^{3} & 1 w^{2} & w^{3} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$
$\Rightarrow (u - v) (v - w) (w - u) (u v + v w + w u) = 0$
given points are distinct. i.e $u \neq v \neq w$
$∴ u v + v w + w u = 0$

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