The length of the latus rectum of the parabola $$x = a{y^2} + by + c$$    is :

Solution :
$$\eqalign{ & x = a\left( {{y^2} + \frac{b}{a}y + \frac{{{b^2}}}{{4{a^2}}}} \right) - \frac{{{b^2}}}{{4a}} + c \cr & {\text{or }}a{\left( {y + \frac{b}{{2a}}} \right)^2} = x + \frac{{{b^2}}}{{4a}} - c \cr & {\text{or }}{\left( {y + \frac{b}{{2a}}} \right)^2} = \frac{1}{a}\left\{ {x + \frac{{{b^2} - 4ac}}{{4a}}} \right\} \cr} $$
$$\therefore $$  the latus rectum $$ = \frac{1}{a}.$$