Class 11

Math

Algebra

Sequences and Series

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

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

$=coff.ofx_{4}−coff.ofx_{2} $

Then,

$p+q+r+s=27 $

And

Arithmetic mean $=4−7/2 =8−7 .$