Generators of the ideal of the projective closure

Question

Let $X=\{(t,t^2,t^5): t\in k\}\subset \mathbb{A}^3$.

1. Show that the projective closure $\bar{X}$ is a projective variety of dimension 1, and say if it is isomorphic to $\mathbb{P}^1$.
2. Compute $\mathbb{I}_p(\bar{X})$.

What I tried:

1. We can see that $X$ is an affine variety with corresponding ideal $J=(y-x^2,z-x^5)$. Since $X$ is an affine variety, then its projective closure is $\mathbb{V}_p({}^hJ)$, where ${}^hJ$ is the homogenization of $J$. We have ${}^h(y-x^2)=wy-x^2$ and ${}^h(z-x^5)=w^4z-x^5.$ Then $\bar{X}=\mathbb{V}_p(wy-x^2,w^4z-x^5)\subset \mathbb{P}^3$ is a projective variety.

Question 1: How can I prove that the dimension of $\bar{X}$ is 1 and decide if it's isomorphic to $\mathbb{P}^1$?

2. To calculate $\mathbb{I}_p(\bar{X})$ I will use that $\mathbb{I}_p(\bar{X})=\mathbb{I}(C(\bar{X}))$, where $C(\bar{X})$ is the cone of $\bar{X}$.

$C(\bar{X})=\{0\} \cup \{x\in\mathbb{A}^4\setminus{\{0\}}: [x]\in\bar{X}\}= \mathbb{V}(wy-x^2,w^4z-x^5)$.

Then $\mathbb{I}(C(\bar{X}))=\sqrt{(wy-x^2,w^4z-x^5)}$.

Question 2. Is correct what I did in 2.?

Question 3. How can I calculate explicitly the radical that I obtained in 2.?