Find equations of the tangents to a parametric curve that pass through a given point Find equations of the tangents to the curve $x=3t^2+1,\space y=2t^3+1$ that pass through the point $(4,3)$


My attempt: 

If we have $x=3t^2+1,\space y=2t^3+1$, then \begin{cases} 4=3t^2+1\\3=2t^3+1 \end{cases}$\implies$\begin{cases} t=1,-1\\ t=1 \end{cases}Therefore we take the parameter of intersection for point $(4,3)$ which is $t=1.$ $$Recall \space \frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{6t^2}{6t}=t$$
So at point $(4,3)$ the slope of the tangent line is $\frac{dy}{dx}=t=1$.

Thus using point slope form: $$y-y_0=\frac{dy}{dx}\big(x-x_0\big)\implies
y-3=x-4\implies y=x-1$$

Our tangent line then is $y=x-1$. However, it can be shown through the elimination of $x$ and $y$ from the tangent line equation that $y=-2x+11$ is also tangent to the curve and passes through $(4,3)$.

My question is:


  How do we know a priori if there exists multiple tangents to a curve through a common point? If it weren't for a solutions manual, I would not think to eliminate $x$ and $y$ to solve for the slope of the tangent. What is the reason there are multiple tangents through a common point? Is this specific to parametric curves?
We can try to solve for it.

$$x=3t^2+1$$
$$y=2t^3+1$$

We have 

$$\frac{dy}{dx}=t$$

Hence the tangent line that we are looking for has the form of 

$$(y-3)=t(x-4)$$

$$(2t^3+1-3)=t(3t^2+1-4)$$
Note that this is a cubic equation, hence we should expect up to $3$ solutions for $t$.

$$(2t^3-2)=t(3t^2-3)$$

$$2(t^3-1)=3t(t^2-1)$$

$$2(t-1)(t^2+t+1)=3t(t-1)(t+1)$$

$$(t-1)(2t^2+2t+2)=(t-1)(3t^2+3t)$$

$$(t-1)(t^2+t-2)=0$$

$$(t-1)(t+2)(t-1)=0$$

$$(t-1)^2(t+2)=0$$

Hence, we would check when $t=1,-2$.
Tangent line definition is a straight line or plane that touches a curve or curved surface at a point, but if extended does not cross it at that point. 
So here even though $(y-3)=-2(x-4)$ satisfies line equation but since it crosses curve at $(4,3)$ its not a tangent line.
There are no multiple tangent lines to the curve which passes through common point