The closure of the set of nondegenerate plane conics that tangent to two fixed lines

Question

Consider a (general) conic in $\mathbb{P}^2$ over the complex number field $\mathbb{C}$, \begin{equation} Q : aX^2+2bXY+cY^2+2dXZ+2eYZ+fZ^2=0. \end{equation} Then we have known that the space of all conics is parametrized by $\mathbb{P}^5$.

Now, I wonder that what is the closure of the Zariski open set that consists of nondegenerate (nonsingular) conics in $\mathbb{P}^3$, that the space of conics tangent to given two projective lines.

At the first glance, I guess the closure is a union of two surfaces isomorphic to $\mathbb{P}^2$ and a quadric $\mathbb{P}^1\times\mathbb{P}^1$.

For example, suppose that $Q$ is tangent to two lines $f_1:(X=0) $ and $f_2:(Y=0) $. Then \begin{equation} Q|_{f_1}: cY^2+2eYZ+fZ^2=0 \end{equation} and then we must have $e^2-cf=0$, while \begin{equation} Q|_{f_2}: aX^2+2dXZ+fZ^2=0 \end{equation} gives $d^2-af=0$.

After we subsituate this consqeuences, we have a set of conics that 1-1 correspondence to a space $\mathbb{P}^3$ that of the forms \begin{equation} Q' : aX^2+2bXY+cY^2+2\sqrt{a}\sqrt{f}XZ+2\sqrt{c}\sqrt{f}YZ+fZ^2=0. \end{equation}

Now, Q is degenerate (singular) conic if and only if the determinant of 3x3 matrix \begin{equation} \left(\begin{array}{ccc} a & b & \sqrt{a}\sqrt{f}\\ b & c & \sqrt{c}\sqrt{f}\\ \sqrt{a}\sqrt{f} & \sqrt{c}\sqrt{f} & f \end{array}\right) \end{equation} is $0$.

1. $(f=0)$ In this case, we always have the determinant $0$, so the variants $b, a, c$ are all free (That is, it fulfills $\mathbb{P}^2$. Furthermore, $Q':aX^2+2bXY+cY^2=0$ and it is either line pair, or double line that pass through $f_1\cap f_2$.

2. $(f\neq 0)$ Then the determinant is $f(b-\sqrt{a}\sqrt{c})$, and the zero set of this formular is a quadric $b-\sqrt{a}\sqrt{c}=0$ in the underlying space $\mathbb{P}^3$. Furthermore, $Q':(\sqrt{a}X+\sqrt{C}Y+\sqrt{f}Z)^2=0$ are double lines meet $f_1$ and $f_2$ separately, namely, $(0:\sqrt{f}:\sqrt{c})$ and $(\sqrt{f}:0:\sqrt{a})$.

Question :

1. Is there any misunderstanding to my claim?

2. Is it true for the general case (for the given any two lines)?

Thank you!