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

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575

576

-576

-575

answer is D.

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Detailed Solution

Given vertex of parabola is (5,4) and directrix is 3x+2-29=0
Co-ordinates of B (foot of directrix)
$\frac{x - 5}{3} = \frac{y - 4}{1} = - (\frac{15 + 4 - 29}{10}) = 1$

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).
$x^{2} + 9 y^{2} - 6 x y + 134 x - 2 y - 2 - 711 = 0$
Compare with given equation. Then a+b+c+d+k=9-6+134+2-711=-576