The equation of the conic section whose focus is at (-1,0) , directrix is the line 4x-3y+2=0 and eccentricity is (1/2)^1/2 is

Hello student,
The eccentricity is given to be √½ < 1. Hence, the given conic section is ellipse.
Now, the focus of the ellipse is F(-1, 0) and directrix is the line 4x-3y + 2 = 0.
Let P(x, y) be any point on ellipse and |MP| be the perpendicular distance from P to the driectrix.
Then, |FP| = e |MP|
So,

This gives 50 (x² + 1 + 2x + y²) = 16x² + 9y² + 4 – 24xy -12y +16x
On simplification, we get
34x² + 41y² + 84x + 12y + 24xy + 46 = 0.