Question

A parabola has a focus of F(−2,−6)and a directrix of x=−4. P(x,y) represents any point on the parabola, while D(−4,y) represents any point on the directrix. Match each numbered step with the correct step in the process of finding the equation of the parabola using the distance formula. 1.FD=FP 2.(x−2)2+(y−6)2=(x−4)2+(y−y)2 3.x2+4x+4+y2+12y+36=x2+8x+16 4.y2−12y+24=4x 5.−14y2−3y+6=x 6.14y2+3y+6=x 7.FP=DP 8. (x−(−2))2+(y−(−6))2=(x−(−4))2+(y−y)2 x2−4x+4+y2−12y+36=x2−8x+16 9. y2+12y+24=4x

(ignore the steps, you decide the steps)

Answer 1

Refer to the figure shown below.

The distance from the focus to the directrix is equal to the distance from the focus to an arbitrary point on the parabola.
Therefore, FD = FP.
FD² = (x = (-4))² = (x+4)²
FP² = (x-(-2))² + (y -(-6))² = (x+2)² + (y+6)²

Obtain
(y+6)² + (x+2)² = (x+4)²
(y+6)² = (x+4)² - (x+2)²
= (x+4+x+2)(x+4-x-2)
= (2x + 6)(2)
= 2x + 12
2x = (y+6)² - 12
x = (1/2)*(y+6)² - 6

Answer:
The equation for the parabola is x = (1/2)*(y+6)² - 6