Q. straight lines cutting cubic... (1 Viewer)

[marsenal]

Find all straight lines y=mx + k which cut the cubic curve y=x³ + Ax² + Bx + c in three equidistant points and show that all these lines pass through a fixed point on the cubic.

Basically I don't even where to start in this question.

 

[spice girl]

Originally posted by marsenal
Find all straight lines y=mx + k which cut the cubic curve y=x³ + Ax² + Bx + c in three equidistant points and show that all these lines pass through a fixed point on the cubic.

Basically I don't even where to start in this question.

Consider the equation x^3 + Ax^2 + (B-m)x + (c-k) = 0

the roots of this equation are the x-coords of the intercepts of the cubic and the line y=mx+k

Now this means that the roots are in arithmetic progression (to be equally spaced in a line). Let these roots be u-d, u, u+d.

sum of roots equal -A
so 3u = -A
u = -A/3 (this is a constant!)

So line must intersect cubic at x=-A/3 (thus pass thru one point).

 

[marsenal]

Thanks! I didn't realise it was a simple as that. I guess the wording stuffed me up.