Question

What are the vertex, focus, and directrix of the parabola with the equation x2+8x+4y+4=0

vertex (4, –3); focus (4, –4); directrix y = –2
vertex (–4, 3); focus (–4, –1); directrix y = 7
vertex (4, –3); focus (4, –7); directrix y = 1
vertex (–4, 3); focus (–4, 2); directrix y = 4

Answer 1

x2 +8x +4y +4 = 0
4y= -x2 -8x -4y = -.25*x^2 -2x -1
a = -.25b = -2c = -1
x position of vertex:
h = -b / 2a
h = 2 / 2*-.25h = 2 / -.5h = -4
y position of vertex:
k = ah^2 + bh + ck = -.25*-4^2 + -2*-4 + -1k = -4 +8 -1k = 3
VERTEX = (-4, 3)**************************************************************************
x value of focus =x value of vertex = -4
y value of focus =(1 (-b^2 -4ac)) / 4a

a = -.25 b = -2 c =-1
y value = (1 (-4 -4*-.25*-1)) / 4*-.25
y value = (1 (-4 -4*-.25*-1)) / -1
y value = (1 -4 +1) / -1y value = (-2 / -1)y value = 2
focus value = (-4, 2)
Answer is the last one.

Answer 2

The Vertex, Focus, Directrix of the given parabola are (-4,3) , (-4,2) and y=4 .

What is a parabola?

A parabola is a curve in which any two points are at the same distance from:

• a fixed point (the focus ), and

• a fixed straight line (the directrix)

x^2 + 8x + 4 = -4y (Rearranging the equation)

(x^2 + 8x + 16) - 12 = -4y

$(x+4)^{2}$ - 12 = -4y

$(x+4)^{2}$ - 12 = -4y

$(x + 4)^{2}$ = 12 - 4y

$(x + 4)^{2}$ = -4(y - 3)

X^2 = -4Y (Changing the vertex of the parabola to (0,0))

X = x + 4 , Y = y - 3

Now this parabola is converted to the parabola of the type X^2 = -4aY.

Focus of X^2 = -4aY is at (0 ,-a)

Hence by comparing we have X = 0 and Y = -a

x + 4 = 0         y - 3 = -1

x = -4               y = 2

Vertex of X^2 = -4aY is at (0,0)

Hence by comparing we have X = 0 and Y = 0

x + 4 = 0          y - 3 = 0

x = -4                y =3    

Equation of directrix of parabola X^2 = -4aY is Y = a.

Hence by comparing we have Y = a

y - 3 = 1

y = 4

Hence, The Vertex, Focus, Directrix of the given parabola are (-4,3) , (-4,2) and y=4 .

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