Find the equation of the tangent to a parabola $\displaystyle y^2=4ax$ at the given pt $\displaystyle (x_1,y_1)$ on it. What can you say about the tangent at $\displaystyle (0,0)$?
Just apply the definition for the first part:
$\displaystyle y - y_1 = \frac{dy}{dx}\bigg{|}_{x = x_1, y = y_1} (x - x_1)$
If $\displaystyle y_1 \neq 0, y - y_1 = \frac{2a}{y_1} (x - x_1)$
$\displaystyle y_1 y - y_1 ^2 = 2ax - 2ax_1$
If $\displaystyle y_1 = 0$, then $\displaystyle x_1 = 0$ and the tangent is the y - axis.