Question

Find coordinate of focus, vertex and equation of directrix and the axis of the parabola y = x² – 2x + 3

Answer 1

The equation of the parabola is x² – 2x + 3 = y

∴ x² – 2x + 1 = y – 2

∴ (x – 1)² = y – 2

Comparing with X² = 4bY, we get

X = x – 1, Y = y – 2, 4b = 1

∴ b = `1/4`

Coordinates of the focus are given by

X = 0, Y = b

∴ x – 1 = 0,  y – 2 = `1/4`

∴ x = 1, y = `9/4`

∴ focus = `(1, 9/4)`

Coordinates of the vertex are X = 0, Y = 0

∴ x – 1 = 0, y – 2 = 0

∴ x = 1, y = 2

∴ vertex = (1, 2)

Equation of directrix is

Y + b = 0

∴ `y - 2 + 1/4` = 0

∴ 4y – 7 = 0

Equation of axis is X = 0

∴ x – 1 = 0, i.e., x = 1.