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A curve is defined by the parametric equation $(x, y, z)$ $=$ $(-2 + 3t, 1 + 3t^2, 2t-3t^3)$. There is a unique point $P$ on the curve with the property that the tangent line at $P$ passes through the point $(-8, 10, 8)$. What are the coordinates of $P$?

So I tried to find the equation of the tangent line first. So what I did was subtract the point $(-8, 10, 8)$ from the general equation of the line. This gave me $(-6-3t, 9-3t^2, 8-2t+3t^3)$. The derivative for the general equation of the line is $(3 , 6t, 2-9t^2)$. So I seem to understand that the tangent line is parallel, and therefore a scalar multiple, of the equation of the derivative. But I can't seem to see what the relationship is and how that can help me get the required coordinates.

Any help?

If the tangent line passes through a fixed point, the cross product of that line and the vector between the fixed point and the tangency point is the zero vector. Thus
$$(-6-3t, 9-3t^2, 8-2t+3t^3)×(3, 6t, 2-9t^2)=(0,0,0)$$
Expanding out the LHS gives a vector whose components are, respectively, a quartic, cubic and quadratic. We only need the quadratic to derive possible values for $t$:
$$-9t^2-36t-27=0$$
$$t^2+4t+3=0qquad t=-1lor t=-3$$
Substituting $t=-3$ into the other components of the cross product does not make them zero, but substituting $t=-1$ does, so it is the correct value and
$$P=(-2 + 3t, 1 + 3t^2, 2t-3t^3)|_t=-1=(-5,4,1)$$

Note that $vecr'=(3,6t,2-9t^2)$ is defined by $vec t=lambdavecr'text, lambda in Bbb R$

So, $$vec P+vec t=(-8,10,8)$$$$ (-2+3t,1+3t^2,2t-3t^3)+(3lambda,6lambda t,2lambda-9lambda t^2)=(-8,10,8)$$

$$implies-2+3t+3lambda=-8$$
$$3(t+lambda)=-6$$
$$t=-2-lambda$$
Similarly, $$1+3t^2+6lambda t=10$$and$$2t-3t^3+2lambda-9lambda t^2=8$$

Can you take it from here?