Question

find the equation of parabola if the vertex is at 2, 1 and the directories is x= Y - 1

Answer 1

vertex of parabola is (2 , 1)
equation of directrix : x - y + 1 = 0

we know, directrix is perpendicular upon axis of parabola .so, Let equation of axis of parabola is x + y + k = 0 [ we know, product of slopes of perpendicular line is -1 ]

we also know, vertex lies on axis of line .
so, (2 , 1) lies on x + y + k =0
so, 2 + 1 + k = 0
k = -3
hence, equation of axis : x + y -3 = 0

now, solve equation of directrix and equation of axis
x = 1 and y = 2
so, (1, 2) is the foot of directrix .
now, Let focus : ( a , b)
we know, vertex is midpoint of foot of directrix and focus .
2 = (a + 1)/2
a =3
1 = (b + 2)/2
b = 0
hence, focus : (3, 0)

now, equation of parabola is
√{(x - 3)² + (y-0)²} = |x - y + 1|/√2
take square both sides,
2(x -3)² +2y² = (x - y + 1)²