Question

Which equation represents a parabola that has a focus of (0 0) and a directrix of y = 2

1. x^2 = - (y-1)
2. x^2 = -4y
3. x^2 = -y
4. x^2 = -4 (y-1)

Answer 1

The answer is 3 because z to the y a-1

Answer 2

4.
`x^2 = -4 (y-1)`

Explanation:

Given,

Focus of parabola = (0,0)

And, directrix is, y = 2,

Thus, the parabola must be along y-axis,

We know that,

The standard form of a parabola along y-axis,

`(x-h)^2=4p(y-k)`

Where, focus = (h, k+p)

And, directrix, y = k-p

Thus, we can write,

h = 0, k+p = 0

k-p = 2

⇒ k+p + k-p = 0 + 2

⇒ 2k = 2 ⇒ k = 1

⇒ 1 - p = 2 ⇒ p = - 1

Hence, the equation of the parabola is,

`(x-0)^2=4(-1)(y-1)`

`\implies x^2 = -4(y-1)`

Option 4 is correct.