Finding a Tangent Point with a Given Slope on a Cubic Curve

[stonecoldgen]

Homework Statement

Find a point P on y=x³ such that the tangent at P intersects the original curve again at point Q so that the slope of the tangent at Q is 4 times the slope of the tangent at P.

Homework Equations

y'=3x²

and the slope of the what I think it should be a secant line=(y2-y1)/(x2-x1)

The Attempt at a Solution

I figured that 2P=Q because 3P²=(3/4)Q² and then by using algebra I expressed Q in terms of P.

I know this leads to somewhere, but I am not sure what should I do next.

 

[LCKurtz]

But P and Q are points. So P² and Q² don't make any sense. Try writing P = (a,a³) and Q= (c,c³) so you have two unknowns a and c. Write the equation of the tangent line at P and use the two facts that it must intersect Q and the slope condition you are given.