Question

homework assignment 8.3 the parabola
score: 5.35/13 answered: 8/13
question 9
write the equation of a parabola whose directrix is ( x = 1 ) and has a focus at ( (-3,9) ).

Explanation:

Step1: Recall parabola definition

For any point $(x,y)$ on parabola, distance to focus = distance to directrix.

Step2: Set up distance equations

Distance to focus $(-3,9)$: $\sqrt{(x+3)^2 + (y-9)^2}$
Distance to directrix $x=1$: $|x-1|$
Equate: $\sqrt{(x+3)^2 + (y-9)^2} = |x-1|$

Step3: Square both sides

$(x+3)^2 + (y-9)^2 = (x-1)^2$

Step4: Expand and simplify

$x^2 +6x +9 + (y-9)^2 = x^2 -2x +1$
Cancel $x^2$: $6x +9 + (y-9)^2 = -2x +1$
Rearrange: $(y-9)^2 = -8x -8$
Factor: $(y-9)^2 = -8(x +1)$

Answer:

$(y-9)^2 = -8(x +1)$