parallel tangent lines?

Question

at what point on the curve x=t^3 +4t, y=6t^2 is the tangent parallel to the line with equations x=-7t, y=12t-5

Answer 1

the curve is: (t^3,6t^2)
the tangent line:
x=-7t
y=-5+12t

the tangent vector to the curve is:
(3t^2,12t)
this is equal to (-7,12)
when
3t^2=-7
and12t=12,
but while the second equation has a solution : t=1
the first equation does not have a solution at all
so the answer is
at no point will the curve be parallel to the given line

Answer 2

Take the derivatives (dx/dt and dy/dt) for each curve. Then set dy/dx = (dy/dt)/(dx/dt) for both curves.
Now set the dy/dx you just got for eath curve equal to each other and solve it for t.


Doug