Mathematics, 04.08.2019 16:30 zookieblu

20 points if its correct will mark brainliest find the standard form of the equation of the parabola with a focus at (0, -2) and a directrix at y = 2. a)y2 = -2x b)y2 = -8x c)y equals negative 1 divided by 8 x squared d)y equals negative 1 divided by 2 x squared find the standard form of the equation of the parabola with a focus at (-8, 0) and a directrix at x = 8. a)y equals negative 1 divided by 32 x squared b)y2 = 16x c)16y = x2 d)x equals negative 1 divided by 32 y squared

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Answer from: jennkoga

For the first parabola we have a focus at (0,-2) and a directrix at y=2. Remember that the distance between any point (x,y)

in the parabola and the focus must be equal to the distance between that point in the parabola and the directrix. Knowing this we will have for a focus at (x_f,y_f)

and a directrix at y_d=c

$\sqrt{(x-x_f)^2+(y-y_f)^2}$
Which is the between a point in the parabola and the focus
|y-c|

Which is the distance from the directrix to a point in the parabola. Then knowing that both must be equal we get:
$\sqrt{(x-x_f)^2+(y-y_f)^2}=|y-c|\implies \sqrt{(x-0)^2+(y+2)^2}=|y-2|$
Applying the squares at both sides and noting that (|x|)^2=x^2

we have:
$(x-0)^2+(y+2)^2=(y-2)^2\implies x^2+y^2+4+4y=y^2+4-4y 
\\
\\
\implies x^2+4y=-4y\implies y=-\frac{x^2}{8}$
Which is the equation of the parabola. The first answer is C).
For the second one we have:
$\sqrt{(x-+8)^2+(y)^2}=|x-8|\implies(x-+8)^2+(y)^2=(x-8)^2$
Simplifying in a similar manner as before we get:
$x=-\frac{y^2}{32}$ The answer is D.
Note that this time the parabola will be sideways with it's vertex on the origin and opening in the negative x direction.

Answer from: Quest

q1 a person or thing situated away or detached from the main body or system. a cluster is a a mash of many things.

q2 do

step-by-step explanation:

Answer from: Quest

the system of inequity is

$y\leq -x+4$

$3y> x$

step-by-step explanation:

from the given graph it is noticed that we have two related lines. one is solid and another is dashed.

if a line passing through two points, then the equation of line is

$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$

the solid line passing through the points (0,4) and (4,0).

$y-4=\frac{0-4}{4-0}(x-0)$

$y=-x+4$

the related equation of solid line is $y=-x+4$ .

similarly, find the related equation of dashed line. the dashed liune passing through the points (0,0) and (3,1).

$y-0=\frac{0-1}{3-0}(x-0)$

$3y=x$

the related equation of dashed line is $3y=x$

from the given graph it is noticed that the point (0,1) lies on the shaded region, therefore the point (0,1) must be satisfied by the related equations.

check the related equations by the point (0,1).

$1=-0+4$

$1=4$

this statement is true if the sign of inequality is < instead of equality. since it is an solid line therefore the sign of inequality will be $\leq$ . the required inequality is

$y\leq -x+4$

similarly, check the second related equation.

$3(1)=0$

it is a dashed line, so this statement is true if the sign of inequality is > . the required inequality is

$3y> x$

15 points write a system of inequalities to represent the shaded portion of the graph

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