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

We need to find the equation of the parabola with vertex $$(5, 4)$$ and directrix $$3x + y - 29 = 0$$.

Find the axis of the parabola

The axis is perpendicular to the directrix. The directrix has slope $$-3$$, so the axis has slope $$\frac{1}{3}$$.

Find the focus

The vertex is the midpoint of the focus and the foot of the perpendicular from the focus to the directrix.

Distance from vertex $$(5, 4)$$ to the directrix $$3x + y - 29 = 0$$:

$$d = \frac{|3(5) + 4 - 29|}{\sqrt{9 + 1}} = \frac{|15 + 4 - 29|}{\sqrt{10}} = \frac{|-10|}{\sqrt{10}} = \frac{10}{\sqrt{10}} = \sqrt{10}$$

So $$a = \sqrt{10}$$ (the distance from vertex to directrix).

The focus is at distance $$a = \sqrt{10}$$ from the vertex on the opposite side of the directrix. The direction from vertex toward the directrix is along $$(3, 1)/\sqrt{10}$$. Since $$3(5) + 1(4) - 29 = -10 < 0$$, the vertex is on the side away from the directrix (toward the origin), so the focus is at:

$$(5, 4) - \sqrt{10} \cdot \frac{(3, 1)}{\sqrt{10}} = (5 - 3, 4 - 1) = (2, 3)$$

Use the definition of a parabola

For any point $$(x, y)$$ on the parabola, the distance to the focus equals the distance to the directrix:

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

Expand

$$10[(x-2)^2 + (y-3)^2] = (3x + y - 29)^2$$

$$10[x^2 - 4x + 4 + y^2 - 6y + 9] = 9x^2 + y^2 + 841 + 6xy - 174x - 58y$$

$$10x^2 - 40x + 40 + 10y^2 - 60y + 90 = 9x^2 + y^2 + 841 + 6xy - 174x - 58y$$

$$10x^2 + 10y^2 - 40x - 60y + 130 = 9x^2 + y^2 + 6xy - 174x - 58y + 841$$

Rearrange to standard form

$$x^2 + 9y^2 - 6xy + 134x - 2y - 711 = 0$$

Compare with the given form

The equation is $$x^2 + ay^2 + bxy + cx + dy + k = 0$$

$$a = 9, \quad b = -6, \quad c = 134, \quad d = -2, \quad k = -711$$

Calculate $$a + b + c + d + k$$

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

Therefore, $$a + b + c + d + k = -576$$.

The correct answer is Option D: $$-576$$.