# Find the equation of the parabola whose focus is (a, b) and whose directrix is x/y + y/b=1

1
by kpankaj022

2015-10-01T22:22:36+05:30

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Directrix D :  x/a + y/b = 1    or  bx + ay -ab = 0
Focus F = (a, b)

let  P(u, v) be the point that satisfies that condition that:
Distance of P from F  = distance of P from D.

(u - a)² + (v - b)² =  (b u + a v - a b)² / (b² + a²)
[ u²+v²+a²+b² - 2a u - 2v b] * (a²+b²) = b²u²+a²v²+a²b²+ 2 ab uv -2 a²vb - 2aub²

u²a² + b²v² + a⁴+a²b²+b⁴ - 2a³ u - 2aub² - 2va²b -2vb³ - 2 abuv + 2a²b v + 2ab² u = 0
a² u² + b² v² -2ab u v - 2u a³ -2 v b³ + (a⁴+b⁴ +a² b²)  = 0
replace  u and v by  x and y:

a² x² + b² y² - 2 ab xy - 2a³ x - 2b³ y + (a⁴ +b⁴ +a²b²) = 0

you can find the axis of the parabola: as it is perpendicular to the directrix.  so:
ax - b y = c
find c  by  knowing that (a, b)  focus lies on the axis.
so  axis:  a x - b y = a² - b²

so  Parabola is:  (ax - by - a²+b²)² = 2a²b² ( x/a + y/b - 3/2)

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