Using a directrix of y = -2 and a focus of (1, 6), what quadratic function is created?

Solution:

Given directrix of y = -2 and focus (1, 6)

From any point (x, y) on the parabola the focus and directrix are equidistant

We are using distance formula √{(x - 1)² + (y - 6)²} = |y + 2|

Applying square on both sides

⇒ (x - 1)² + (y - 6)² = (y + 2)²

⇒ (y - 6)² - (y + 2)² = -(x - 1)²

⇒ y² - 12y + 36 - y² - 4y - 4 = -(x - 1)²

⇒ -16y + 32 = -(x - 1)²

⇒ -16y = -(x - 1)² - 32

⇒ y = (x - 1)² /16 + 32/16

⇒ y = (x - 1)² /16 + 2

 The quadratic equation created is y= (x - 1)² /16 + 2.

Using a directrix of y = -2 and a focus of (1, 6), what quadratic function is created?

Summary:

Using a directrix of y = -2 and a focus of (1, 6), y = (x - 1)² /16 + 2 quadratic function is created.