## Calculus (3rd Edition)

$$2tx+ty+cz=d.$$
Let the equation of any plane that intersects the $xy$-plane in the line $r(t)=t\langle 2,1,0\rangle$ be given by $$ax+by+cz=d.$$ Now, to find the $xy$ intersection, we put $z=0$ and hence we get $$ax+ by=d.$$ Due to the intersection, we have $$a=2t, \quad b=t .$$ So the equation of any plane whose intersection with the $xy$-plane is the line $r(t)=t\langle 2,1,0\rangle$ is given by $$2tx+ty+cz=d.$$