Three distinct points $P (3 u^{2}, 2 u^{3}); Q (3 v^{2}, 2 v^{3}) a n d R (3 w^{2}, 2 w^{3})$ are collinear and equation $a x^{3} + b x^{2} + c x + d = 0$ has roots u, v and w, then which of the following is true

b = 0

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c = 0

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d = 0

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ab = 0

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Solution

The correct option is B
c = 0

$∣ ∣ ∣ ∣ ∣ \begin{matrix} 3 u^{2} & 2 u^{3} & 1 3 v^{2} & 2 v^{3} & 1 3 w^{2} & 2 w^{3} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0 \Rightarrow 6 ∣ ∣ ∣ ∣ ∣ \begin{matrix} u^{2} & u^{3} & 1 v^{2} & v^{3} & 1 w^{2} & w^{3} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$
$\Rightarrow$ (u - v)(v - w)(w - u)(uv + vw + wu) = 0
$\Rightarrow$ uv + vw + wu = 0 (since u, v, w are distinct)
$\Rightarrow$ sum of roots product of roots taken two at a time
= uv + vw + wu = 0 = $\frac{c}{a}$
$\Rightarrow$ c = 0

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