Question

homework assignment 8.3 the parabola
score: 3.6/13 answered: 3/13
question 4
find the equation for the parabola that has its focus at $(-1,0)$ and has directrix $y = 6$.
the equation is:
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Explanation:

Step1: Recall parabola definition

A parabola is the set of points $(x,y)$ equidistant from focus $(h,k+p)$ and directrix $y=k-p$. Given focus $(-1,0)$ and directrix $y=6$.

Step2: Find vertex $(h,k)$

Vertex is midpoint of focus and directrix projection:
$h=-1$, $k=\frac{0+6}{2}=3$.

Step3: Calculate $p$

Distance from vertex to focus: $p=0-3=-3$.

Step4: Write standard equation

Vertical parabola: $(x-h)^2=4p(y-k)$
Substitute $h=-1,k=3,p=-3$:
$(x+1)^2=4(-3)(y-3)=-12(y-3)$

Step5: Expand to general form

$(x^2 + 2x +1) = -12y +36$
Rearrange: $x^2 +2x +12y -35=0$

Answer:

$x^2 + 2x + 12y - 35 = 0$ (or $(x+1)^2 = -12(y-3)$)