Find the equation

Given that we need to find the equation of the ellipse whose focus is S(1, - 2) and directrix(M) is 3x - 2y + 5 = 0 and eccentricity(e) is equal to .

Let P(x,y) be any point on the ellipse.

We know that the distance between the focus and any point on the ellipse is equal to the eccentricity times the perpendicular distance from that point to the directrix.

We know that distance between the points (x₁,y₁) and (x₂,y₂) is .

We know that the perpendicular distance from the point (x₁,y₁) to the line ax + by + c = 0 is .

⇒ SP = ePM

⇒ SP² = e²PM²

⇒

⇒

⇒

⇒ 52x² + 52y² - 104x + 208y + 260 = 9x² + 4y² - 12xy - 20y + 30x + 25

⇒ 43x² + 48y² + 12xy - 134x + 228y + 235 = 0

∴ The equation of the ellipse is 43x² + 48y² + 12xy - 134x + 228y + 235 = 0.