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

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by kpankaj022

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by kpankaj022

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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)

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:

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)