The length of the latus rectum of the parabola $$169\left\{ {{{\left( {x - 1} \right)}^2} + {{\left( {y - 3} \right)}^2}} \right\} = {\left( {5x - 12y + 17} \right)^2}$$         is :

Solution :
Here, $${\left( {x - 1} \right)^2} + {\left( {y - 3} \right)^2} = {\left\{ {\frac{{5x - 12y + 17}}{{\sqrt {{5^2} + {{\left( { - 12} \right)}^2}} }}} \right\}^2}$$
$$\therefore $$  the focus $$ = \left( {1,\,3} \right)$$  and the directrix is $$5x - 12y + 17 = 0$$
The distance of the focus from the directrix $$ = \left| {\frac{{5 \times 1 - 12 \times 3 + 17}}{{\sqrt {{5^2} + {{\left( { - 12} \right)}^2}} }}} \right| = \frac{{14}}{{13}}$$
$$\therefore $$  latus rectum $$ = 2 \times \frac{{14}}{{13}} = \frac{{28}}{{13}}.$$