Question

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

• A. b = 0
• B. c = 0
• C. d = 0
• D. ab = 0

Answer 1

The correct option is B

c = 0

$\left|\begin{array}{ccc}3u^2& 2u^3& 1\\ 3v^2& 2v^3& 1\\ 3w^2& 2w^3& 1\end{array}\right|=0\Rightarrow 6\left|\begin{array}{ccc}{u}^2& u^3& 1\\ v^2& v^3& 1\\ w^2& w^3& 1\end{array}\right|=0$
$\Rightarrow$ (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