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If the distinct points on the curve y = 2{x^4} + 7{x^3} + 3x -

Question

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

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If the distinct points on the curve

y = 2 x^{4} + 7 x^{3} + 3 x - 5

are collinear, then find the arithmatic mean of x-coordinates of the aforesaid points

10 mins ago

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10 mins ago

Text solutionVerified

Given ; Four distinct points on the curve.

y = 2 x^{4} + 7 x^{3} + 3 x - 5

are collinear

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

so,

m x + c = 2 x^{4} + 7 x^{3} + 3 x - 5

2 x^{4} + 7 x^{3} + 3 x - 5 - m x - c = 0

2 x^{4} + 7 x^{3} + 3 x - m x - 5 - c = 0

2 x^{4} + 7 x^{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{- co ff . o f x ^{2}}{co ff . o f x ^{4}}

Then,

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

And

Arithmetic mean

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

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Question Text	If the distinct points on the curve $y = 2 x^{4} + 7 x^{3} + 3 x - 5$ are collinear, then find the arithmatic mean of x-coordinates of the aforesaid points
Answer Type	Text solution:1
Upvotes	150