If the vertex of a parabola is $\left(2,-1\right)$ And the equation of its directrix is $4x-3y=21,$ then the length of its latus rectum  

For any parabola, If the distance between the vertex and directrix is $a$ then the length of the latusrectum is $4a$

The perpendicular distance from the vertex $\left(2,-1\right)$ to the directrix $4x-3y=21,$ is $a$

Hence, $a=\left|\frac{8+3-21}{\sqrt{16+9}}\right|=\left|\frac{10}{5}\right|=2$

Therefore, the length of the latusrectum is $4a=8$