In $R^3,$ the two surfaces with the Cartesian equations $x=4\sqrt[\phantom{2}]{y^2+9}$ and $2xy-9z=0$ intersect to form a curve $C,$ which passes through the point $P$:$(x,y,z)=(12,0,0).$ The curve $C$ is parametrized by the function $r$:$R{R}^3$ that is defined by $r(t)=12cosh(t)i+3sinh(t)j+4sinh(2t)k.$

a$)$ Verify for all tinR that $r(t)$ does indeed lie on the intersection of the two surfaces. $(4$ marks$)$

b$)$ Determine a vector equation of the line $L$ that is tangent to the curve $C$ at the point $P.(6$ marks$)$