Let P(x, y) be any point on the parabola whose focus is S(– 1, – 2) and the directrix x – 2y + 3 = 0.
Draw PM perpendicular to directrix x – 2y + 3 = 0. Then by definition,
SP = PM
⇒ SP2 = PM2
⇒ 5 [(x + 1)2 + (y + 2)2 ] = (x – 2y + 3)2
⇒ 5(x2 + y2 + 2x + 4y + 5) = (x2 + 4y2 + 9 – 4xy + 6x – 12y)
⇒ 4x2 + y2 + 4xy + 4x + 32y + 16 = 0
This is the equation of the required parabola.