Question

Find the focus of the parabola that has a vertex at (0,0) and that passes through the points (-3,3) and (3,3)

Answer 1

Focus = (0,
`(3)/(4)`)

Explanation:

(± 3, 3) are at an equal distance from y-axis.

axis of parabola = y-axis

vertex = (0, 0)

Parabola will be of the form: x² = 4ay, passing through(± 3, 3)

(± 3)² = 4 × a × 3 ⇒ 9 = 12a ⇒ a =
`(9)/(12)`

a =
`(3)/(4)`

Coordinates of focus are: (0, a) ⇒ (0,
`(3)/(4)`)

Answer 2

The focus of the parabola is (0 , 3/4)

Explanation:

* Lets revise some facts about the parabola

- The standard form of the equation of a parabola of vertex (h , k)

is (x - h)² = 4p (y - k)

- The standard form of the equation of a parabola of vertex (0 , 0) is

x² = 4p y, from this equation we can find:

# The axis of symmetry is the y-axis, x = 0

# 4p equal to the coefficient of y in the given equation

# If p > 0, the parabola opens up.

# If p < 0, the parabola opens down.

# The coordinates of the focus, (0 , p)

# The directrix , y = − p

* Now lets solve the problem

∵ The vertex of the parabola is (0 , 0)

∴ The equation of the parabola is x² = 4p y

∵ the parabola passes through points (-3 , 3) and (3 , 3)

- Substitute the value of x and y coordinates of one point in the

equation to find the value of p

∴ (3)² = 4p (3) ⇒ we use point (3 , 3)

∴ 9 = 12 p ⇒ divide both sides by 12

∴ p = 9/12 = 3/4

- Now lets find the focus of the parabola

∵ The focus of the parabola is (0 , p)

∵ p = 3/4

∴ The focus of the parabola is (0 , 3/4)