See , in general if the eqn of directric of a parabola is ax+by+c=0 and the co-ordinates of the focus are (h,k) . Then the locus of the pt such that it's distance from the directrix and the focus is equal would give the rqd parabola.

Let the pt whose locus is to be found be (h',k')

Then,by definition of a parabola

(h'-h)² + (k'-k)² = (ah'+bk'+c)² / a²+b²

 

put x,y instead of h',k' to get the locus which would be

 

=> (a²+b²)((x-h)² + (y-k)²) = (ax+by+c)²

 

Now for any parabola of this form the eqn of directrix is ax+by+c = 0 and the focus is (h,k)...this is called the general eqn of the parabola.

 

Now , your qn seems fairly simple.

Eqn of directrix = 5x-12y+17 = 0

and focus is (1,3)

The distance b/w the focus and the directrix = 2a = 14/13 in this case

Hence length of latus rectum is 4a = 28/13.....(ans)