Question

If the distinct points on the curve $y=2x^4+7x^3+3x-5$ are collinear, then find the arithmatic mean of x-coordinates of the aforesaid points

Answer 1

Given ; Four distinct points on the curve.

$y=2x^4+7x^3+3x-5$ are collinear

Let all points are on y = mx + c (That is general from of linear equation

so,

$mx+c=2x^4+7x^3+3x-5$

$2x^4+7x^3+3x-5-mx-c=0$

$2x^4+7x^3+3x-mx-5-c=0$

$2x^4+7x^3+(3-m)x-(5+c)=0$

Let that equation have four roots p,q,r & s

And we know sum of roots in bi-quadratic equation

$=\frac{-coff.of{x}^2}{coff.of{x}^4}$

Then,

$p+q+r+s=\frac{7}{2}$

And

Arithmetic mean $=\frac{-7/2}{4}=\frac{-7}{8}.$