If the equation of the parabola, whose vertex is at $(5,4)$ and the directrix is $3x+y-29=0$, is $x^2+a{y}^2+bxy+cx+dy+k=0$ then $a+b+c+d+k$ is equal to

Complete Solution:

Given Data:

Vertex: (5, 4)

Directrix: 3x + y - 29 = 0

General Form of the Parabola:

The parabola is given in the form: x² + a y² + b x y + c x + d y + k = 0

After applying the conditions of the vertex and directrix and simplifying the equation, we find: a + b + c + d + k = -576

Final Answer:

The value of a + b + c + d + k is -576.