Which equation represents a parabola with a focus of (0, 4) and a directrix of y = 2?
y = x² + 3
y = -x² + 1
y = x²/2 + 3
y = x²/4 + 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 = √(x₂ - x₁)² + (y₂ - y₁)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

⇒ x² + y² - 8y + 16 = y² - 4y + 4

Grouping of common terms,

⇒ x² + y² - y² - 8y + 4y + 16 - 4 = 0

⇒ x² - 4y + 12 = 0

⇒ x² = 4y - 12

⇒ 4y = x² + 12

⇒ y = 1/4[x² + 12]

⇒ y = (x²/4) + 3

Therefore, the quadratic function is y = (x²/4) + 3.

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

Summary:

The equation y = (x²/4) + 3 represents a parabola with a focus of (0, 4) and a directrix of y = 2.