I am trying to calculate the tangent cone of $\mathbb{A}^1$ at the point $Q=1$. To do that, I first want to calculate the tangent cone at the origin, and then "translate back".
Here is the definition that I use. For an affine variety $V= \text{Specm }k[X_1,\ldots X_n]/\mathfrak{a}$ over an algebraically closed field $k$, we can define the ideal $\mathfrak{a}_* =\{f_*:f\in\mathfrak{a}\}$ where, for $f\in k[X_1,\ldots X_n]$, $f_*$ is the homogeneous part of $f$ of lowest degree. The tangent cone at the origin is defined as the $k$-algebra $k[X_1,\ldots,X_n]/\mathfrak{a}_*$.
I know that $\mathbb{A}^1$ is defined by the polynomial $0$. Does this not mean that $\mathfrak{a}_*=(0)$? Then we would have that the tangent cone at the origin is equal to $k[X]$.
If this is the tangent cone at the origin, then something tells me that the tangent cone at $Q$ must be $k[Y]/(Y-1)$. However, I don't entirely see why. Any help is greatly appreciated!