The equation of the ellipse whose focus is $(3, - 2)$ , eccentricity $\frac{3}{4}$ and directrix $2 x - y + 3 = 0$ is

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$14 x^{2} + 33 x y + 17 y^{2} - 225 x + 74 y + 159 = 0$

$44 x^{2} - 36 x y - 71 y^{2} - 588 x - 374 y + 959 = 0$

$44 x^{2} + 36 x y + 71 y^{2} - 588 x + 374 y + 959 = 0$

$4 x^{2} + 56 x y + 271 y^{2} - 188 x + 274 y + 359 = 0$

answer is B.

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Detailed Solution

Let $P (x, y)$ be the locus of the point

Equation of conic $S P^{2} = e^{2} P M^{2}$

${(x - 3)}^{2} + {(y + 2)}^{2} = \frac{9}{16} {| \frac{2 x - y + 3}{\sqrt{5}} |}^{2} \Rightarrow x^{2} + y^{2} - 6 x + 4 y + 13 = \frac{9}{16} (\frac{4 x^{2} + y^{2} + 9 - 4 x y - 6 y + 12 x}{5}) 80 x^{2} + 80 y^{2} - 480 x + 320 y + 1040 = 36 x^{2} + 9 y^{2} + 81 - 36 x y - 54 y + 108 x 44 x^{2} + 71 y^{2} + 36 x y - 588 x + 374 y + 959 = 0$