Fix a curve $C$ and a line both in $\mathbb{P}^3$ not intersecting one another. Suppose that the ideal of the curve is generated by $P_i$ and the line is definied by linear polynomials $l_1, l_2$. Is it true that a general degree $d$ smooth surface in $\mathbb{P}^3$ containing $C \cup l$ is defined by an equation of the form $$l_1\left(\sum_iP_iQ_i\right)+l_2\left(\sum_iP_iQ'_i\right)$$ for $Q_i, Q'_i$ homogeneous polynomials of degree equal to $d-1-\deg(P_i)$?

Fix a curve $C$ and a line both in $\mathbb{P}^3$ not intersecting one another. Suppose that the ideal of the curve is generated by $P_i$ and the line is definied by linear polynomials $l_1, l_2$. Is it true that a general degree $d$ smooth surface in $\mathbb{P}^3$ containing $C \cup l$ is defined by an equation of the form $$ l_1\left(\sum_iP_iQ_i\right)+l_2\left(\sum_iP_iQ'_i\right) $$ for $Q_i, Q'_i$ homogeneous polynomials of degree equal to $d-1-\deg(P_i)$?

Fix a curve $C$ and a line both in $\mathbb{P}^3$ not intersecting one another. Suppose that the ideal of the curve is generated by $P_i$ and the line is definied by linear polynomials $l_1, l_2$. Is it true that a general degree $d$ smooth surface in $\mathbb{P}^3$ containing $C \cup l$ is defined by an equation of the form

     $l_1(\sum_iP_iQ_i)+l_2(\sum_iP_iQ'_i)$

for $Q_i, Q'_i$ homogeneous polynomials of degree equal to $d-1-\deg(P_i)$?

Fix a curve $C$ and a line both in $\mathbb{P}^3$ not intersecting one another. Suppose that the ideal of the curve is generated by $P_i$ and the line is definied by linear polynomials $l_1, l_2$. Is it true that a general degree $d$ smooth surface in $\mathbb{P}^3$ containing $C \cup l$ is defined by an equation of the form $$l_1\left(\sum_iP_iQ_i\right)+l_2\left(\sum_iP_iQ'_i\right)$$ for $Q_i, Q'_i$ homogeneous polynomials of degree equal to $d-1-\deg(P_i)$?