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

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575

-575

576

-576

answer is D.

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Detailed Solution

Given vertex of parabola is (5, 4) and directrix is $3 x + y - 29 = 0$
Co-ordinates of B (foot of directrix)
$\frac{x - 5}{3} = \frac{y - 4}{1} = - (\frac{15 + 4 - 29}{10}) = 1$
Question Image
X = 8, y = 5
So, focus will be at same distance to vertex as directrix.
Then, F = (2, 3) (focus)
Equation of parabola is PF = PM with point P(x, y).
$x^{2} + 9 y^{2} - 6 x y + 134 x - 2 y - 711 = 0$
Compare with given equation. Then,
$a + b + c + d + k = 9 - 6 + 134 - 2 - 711 = - 576$