show, slope of the tangent at Q is four times the slope atP

[wind]

Suppose that the tanent at a point P on the curve y=x^3 intersects the curve again at a point Q. Show that the slope of the tangent at Q is four times the slope of the tangent at P.

y= x^3
y'=3x^2

mq=4mp

I don't get it... can someone pleas help? Thanks

 

[skeeter]

let P have coordinates (p, p^3)

let Q have coordinates (q, q^3)

tangent line at P passes thru Q ...

slope of tangent line at P is 3p^2

it is also (p^3 - q^3)/(p - q) = p^2 + pq + q^2

p^2 + pq + q^2 = 3p^2

q^2 + pq - 2p^2 = 0

(q - p)(q + 2p) = 0

q cannot equal p since y = x^3 has no vertical tangent lines, so
q = -2p

so ... slope of the tangent line at Q is 3q^2 = 3(-2p)^2 = 12p^2, which is 4 times the slope of the tangent line at P.

 

[wind]

Thanks, skeeter.