# Which equation represents a parabola with a focus of (0, 4) and a directrix of y = 2?

y = x^{2} + 3

y = -x^{2} + 1

y = x^{2}/2 + 3

y = x^{2}/4 + 3

**Solution:**

The definition of a parabola states that all points on the parabola always have the same distance to the focus and the directrix.

Let A = (x, y) be a point on the parabola.

Focus, F = (0, 4)

Given, directrix y = 2

D = (x, 2) represent the closest point on the directrix

First, find out the distance using the distance formula,

d = √(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Distance A and F is dAF = √(x - 0)^{2} + (y - 4)^{2} = √(x)^{2} + (y - 4)^{2}

Distance between A and D is dAD = √(x - x)^{2} + (y - 2)^{2} = √(y - 2)^{2}

Since these distances must be equal to each other,

⇒ √(x)^{2} + (y - 4)^{2} = √(y - 2)^{2}

Squaring both sides,

⇒ (√(x)^{2} + (y - 4)^{2})^{2} = (√(y - 2)^{2})^{2}

⇒ (x)^{2} + (y - 4)^{2} = (y - 2)^{2}

⇒ x^{2} + y^{2} - 8y + 16 = y^{2} - 4y + 4

Grouping of common terms,

⇒ x^{2} + y^{2} - y^{2} - 8y + 4y + 16 - 4 = 0

⇒ x^{2} - 4y + 12 = 0

⇒ x^{2 }= 4y - 12

⇒ 4y = x^{2 }+ 12

⇒ y = 1/4[x^{2} + 12]

⇒ y = (x^{2}/4) + 3

Therefore, the quadratic function is y = (x^{2}/4) + 3.

## Which equation represents a parabola with a focus of (0, 4) and a directrix of y = 2?

**Summary:**

The equation y = (x^{2}/4) + 3 represents a parabola with a focus of (0, 4) and a directrix of y = 2.

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