The focus of a parabola is $S(-1,1)$ and the equation of the directrix is $4x+3y-24=0$, then

The equation of the axis is perpendicular to the directrix then slope of

$4x+3y-24=0$ is  $m=\frac{-4}{3}$

Slope of axis is $\frac{3}{4}$

S $(-1,1)$ equation of the axis is

$y-1=\frac{3}{4}(x+1)\Rightarrow 3x-4y+7=0$

Vertex is midpoint of S and Z

$Z(3,4),S(-1,1)$

$A=\left(\frac{3-1}{2},\frac{4+1}{2}\right)=\left(1,\frac{5}{2}\right)$

$\therefore A=(1,\frac{5}{2})$

Tangent at vertex $(1,\frac{5}{2})$

Clearly the tangent at vertex is $y-\frac{5}{2}=\frac{-4}{3}(x-1)$

$8x+6y-23=0$

Length of the latus rectum $=2\times 5=10$