Question

Derive the equation of the parabola with a focus at (0, -4) and a directrix of y=4

Answer 1

Answer:

C

Step-by-step explanation:

The focus and directrix are equidistant from any point (x, y) on the parabola.

Using the distance formula

$\sqrt{(x-0)^2+(y+4)^2}$ = | y - 4 |

$\sqrt{x^2+(y+4)^2}$ = | y - 4 |

Square both sides

x² + (y + 4)² = (y - 4)² ← expand parenthesis on both sides

x² + y² + 8y + 16 = y² - 8y + 16

Subtract y² - 8y + 16 from both sides

x² + 16y = 0 ( subtract x² from both sides )

16y = - x² ( divide both sides by 16 )

y = - $\frac{1}{16}$ x², or

f(x) = - $\frac{1}{16}$x² → C

Answer 2

There are several parameters that can be used to determine the equation of a parabola

The equation of the parabola i $\mathbf{y =-\frac{x^2}{16}}$

The given parameters are given as:

$\mathbf{(a,b) = (0,-4)}$ ---- focus

$\mathbf{y =4}$ --- directrix

The equation of the parabola is calculated using:

$\mathbf{\sqrt{(x - a)^2 + (y - b)^2} = |y - directrix|}$

So, we have:

$\mathbf{\sqrt{(x - 0)^2 + (y + 4)^2} = |y - 4|}$

$\mathbf{\sqrt{x^2 + (y + 4)^2} = |y - 4|}$

Square both sides

$\mathbf{x^2 + (y + 4)^2 = (y - 4)^2}$

Evaluate all exponents

$\mathbf{x^2 + y^2 + 8y + 16 = y^2 - 8y + 16}$

Cancel out common terms

$\mathbf{x^2 + 8y =- 8y}$

Collect like terms

$\mathbf{x^2 + 8y+8y =0}$

$\mathbf{x^2 + 16y =0}$

Make y the subject

$\mathbf{16y =-x^2}$

Divide both sides by 16

$\mathbf{y =-\frac{x^2}{16}}$

Hence, the equation of the parabola i $\mathbf{y =-\frac{x^2}{16}}$

Read more about parabolas at:

https://brainly.com/question/4074088