JEE Main · 2022 · 27 Jun 2022 · Shift 2 · Parabola

The vertex is $(5,4)$ and directrix is $3x+y-29=0$. The focus lies on the perpendicular from the vertex to the directrix.

The foot of perpendicular from $(5,4)$ to the directrix:
$\frac{x-5}{3}=\frac{y-4}{1}=-\frac{15+4-29}{10}=1$

So the foot is at $(8,5)$. The focus is at $(2,3)$ (reflection of vertex about foot).

The parabola equation is: distance from point to focus = distance from point to directrix.

$(x-2{)}^2+(y-3{)}^2=\frac{(3x+y-29{)}^2}{10}$

Expanding and simplifying:
$x^2+9y^2-6xy+134x-2y-711=0$

Comparing: $a=9,b=-6,c=134,d=-2,k=-711$

$a+b+c+d+k=9-6+134-2-711=-576$

Answer: (D)