Find co-ordinates of focus, vertex, and equation of directrix and the

1.	Find co-ordinates of focus, vertex, and equation of directrix and the axis of the parabola y = x2 – 2x + 3
Answer» Given equation of the parabola is y = x² – 2x + 3 ⇒ y = x – 2x + 1 + 2 ⇒ y – 2 = (x – 1)² ⇒ (x – 1)² = y – 2 Comparing this equation with X² = 4bY, we get X = x – 1, Y = y – 2 ⇒ 4b = 1 ⇒ b = 1/4 The co-ordinates of vertex are (X = 0, Y = 0) ⇒ x – 1 = 0 and y – 2 = 0 ⇒ x = 1 and y = 2 The co-ordinates of vertex are (1, 2). The co-ordinates of focus are S(X = 0, Y = b) ⇒ x – 1 = 0 and y – 2 = 1/4 ⇒ x = 1 and y = 9/4 The co-ordinates of focus are (1, 9/4) Equation of the axis is X = 0 x – 1 = 0, i.e., x = 1 Equation of directrix is Y + b = 0 ⇒ y – 2 + 1/4 =0 ⇒ y – 7/4 = 0 ⇒ 4y – 7 = 0

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

Answer»

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

⇒ y = x – 2x + 1 + 2

⇒ y – 2 = (x – 1)²

⇒ (x – 1)² = y – 2

Comparing this equation with X² = 4bY, we get

X = x – 1, Y = y – 2

⇒ 4b = 1

⇒ b = 1/4

The co-ordinates of vertex are (X = 0, Y = 0)

⇒ x – 1 = 0 and y – 2 = 0

⇒ x = 1 and y = 2

The co-ordinates of vertex are (1, 2).

The co-ordinates of focus are S(X = 0, Y = b)

⇒ x – 1 = 0 and y – 2 = 1/4

⇒ x = 1 and y = 9/4

The co-ordinates of focus are (1, 9/4)

Equation of the axis is X = 0

x – 1 = 0, i.e., x = 1

Equation of directrix is Y + b = 0

⇒ y – 2 + 1/4 =0

⇒ y – 7/4 = 0

⇒ 4y – 7 = 0