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

Given: Vertex of the parabola is $$(2, -1)$$ and the equation of the directrix is $$4x - 3y = 21$$.

Find the distance from the vertex to the directrix:

The perpendicular distance from a point $$(x_0, y_0)$$ to the line $$ax + by + c = 0$$ is:

$$d = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}}$$

Rewriting the directrix as $$4x - 3y - 21 = 0$$:

$$d = \frac{|4(2) - 3(-1) - 21|}{\sqrt{16 + 9}} = \frac{|8 + 3 - 21|}{5} = \frac{|-10|}{5} = 2$$

Relate the distance to the latus rectum:

For a parabola, the distance from the vertex to the directrix equals $$a$$ (the focal distance). Therefore $$a = 2$$.

Find the length of the latus rectum:

$$\text{Length of latus rectum} = 4a = 4 \times 2 = 8$$

The correct answer is Option B: $$8$$.