Question

find the vertex, focus, and directrix for the following parabolas.
(a)
$(y - 9)^2 = 12(x - 5)$
vertex :
focus :
directrix:
(b)
$y^2 - 10y = 12x - 25$
vertex :
focus :
directrix:
(c)
$(x - 4)^2 = 12(y - 3)$
vertex :
focus :
directrix:
(d)
$x^2 + 24x = 4y - 36$
vertex :
focus :
directrix:

Explanation:

Part (a)

Step1: Identify vertex from standard form

Standard form: $(y - k)^2 = 4p(x - h)$. Here, $h=5$, $k=9$. Vertex: $(h,k)$.

Step2: Calculate $p$

$4p=12 \Rightarrow p=3$.

Step3: Find focus

Focus: $(h+p,k)=(5+3,9)$.

Step4: Find directrix

Directrix: $x=h-p=5-3$.

Part (b)

Step1: Complete the square for $y$

$y^2 -10y = (y-5)^2 -25$. Equation becomes $(y-5)^2 -25 =12x -25 \Rightarrow (y-5)^2=12x$.

Step2: Identify vertex

$h=0$, $k=5$. Vertex: $(0,5)$.

Step3: Calculate $p$

$4p=12 \Rightarrow p=3$.

Step4: Find focus

Focus: $(h+p,k)=(0+3,5)$.

Step5: Find directrix

Directrix: $x=h-p=0-3$.

Part (c)

Step1: Identify vertex from standard form

Standard form: $(x - h)^2 =4p(y - k)$. Here, $h=4$, $k=3$. Vertex: $(4,3)$.

Step2: Calculate $p$

$4p=12 \Rightarrow p=3$.

Step3: Find focus

Focus: $(h,k+p)=(4,3+3)$.

Step4: Find directrix

Directrix: $y=k-p=3-3$.

Part (d)

Step1: Complete the square for $x$

$x^2 +24x=(x+12)^2 -144$. Equation becomes $(x+12)^2 -144=4y -36 \Rightarrow (x+12)^2=4y +108 \Rightarrow (x+12)^2=4(y +27)$.

Step2: Identify vertex

$h=-12$, $k=-27$. Vertex: $(-12,-27)$.

Step3: Calculate $p$

$4p=4 \Rightarrow p=1$.

Step4: Find focus

Focus: $(h,k+p)=(-12,-27+1)$.

Step5: Find directrix

Directrix: $y=k-p=-27-1$.

Answer:

(a) vertex: $(5,9)$; focus: $(8,9)$; directrix: $x=2$
(b) vertex: $(0,5)$; focus: $(3,5)$; directrix: $x=-3$
(c) vertex: $(4,3)$; focus: $(4,6)$; directrix: $y=0$
(d) vertex: $(-12,-27)$; focus: $(-12,-26)$; directrix: $y=-28$