What is the answer and how did you get the answer

Find the focus and the directrix for the parabola with the given equation.

musickatia [10]

Answer:

$\mathrm{Focus:\;\;} \left(-\frac{5}{3},\:2\right)$

$\mathrm{Directrix:\;\;}x=-\frac{1}{3}$

Step-by-step explanation:

The standard equation for a parabola is

$4p\left(x-h\right)=\left(y-k\right)^2$

with vertex at (h, k) and a focal length of |p|

$\mathrm{Rewrite}\:x=-\frac{3}{8}\left(y-2\right)^2-1\:\mathrm{in\:the\:standard\:form}$ :

$\mathrm{Add\:}1\mathrm{\:to\:both\:sides}$

$x+1=-\frac{3}{8}\left(y-2\right)^2-1+1$ or

$x+1=-\frac{3}{8}\left(y-2\right)^2$

$\mathrm{Divide\:both\:sides\:by\:}-\frac{3}{8}$
$\frac{x+1}{-\frac{3}{8}}=\frac{-\frac{3}{8}\left(y-2\right)^2}{-\frac{3}{8}}$

$\mathrm{Simplify}$
$-\frac{8x}{3}-\frac{8}{3}=\left(y-2\right)^2$

$\mathrm{Factor\:}-\frac{8}{3}$
$\left(-\frac{8}{3}\right)\left(x+\frac{-\frac{8}{3}}{-\frac{8}{3}}\right)=\left(y-2\right)^2$

$\mathrm{Simplify}$
$\left(-\frac{8}{3}\right)\left(x+1\right)=\left(y-2\right)^2$

$\mathrm{Factor\:}4$
$4\cdot \frac{-\frac{8}{3}}{4}\left(x+1\right)=\left(y-2\right)^2$

$\mathrm{Simplify}$
$4\left(-\frac{2}{3}\right)\left(x+1\right)=\left(y-2\right)^2$

$\mathrm{We\; can\; rewrite\:this\;as}$
$4\left(-\frac{2}{3}\right)\left(x-\left(-1\right)\right)=\left(y-2\right)^2$

Comparing this with the standard form we get
$\left(h,\:k\right)=\left(-1,\:2\right)$
$\:p=-\frac{2}{3}$

The parabola is symmetric around the x-axis.

The focus lies a distance $p$ from the center $\left(-1,\:2\right)$ along the x axis

So focus is at
$\left(-1+p,\:2\right)$ $=\left(-1+\left(-\frac{2}{3}\right),\:2\right)$ $=\bold{ \left(-\frac{5}{3},\:2\right)}$

The parabola is symmetric around the x-axis and so the directrix is a line parallel to the y-axis at a distance -p from the center

$x=-1-p$

$x=-1-\left(-\frac{2}{3}\right) = \bold{-\frac{1}{3}}$

3 0

2 years ago

Help please <br> Rewrite the equation −4.9t^2+6.1t+2.2=3 in standard form.

Mnenie [13.5K]

COuld you send a picture

4 0

4 years ago

Max and Sasha exercise a total of 20 hours each week. Max exercises 15 hours less than 4 times the

navik [9.2K]

Answer:

Max exercises 13 hours a week, and Sasha 7.

Step-by-step explanation:

To find the number of hours each of them exercises during the week, we solve the system of equations.

In the second equation:

$x = 4y - 15$

Replacing in the first equation:

$x + y = 20$

$4y - 15 + y = 20$

$5y = 35$

$y = \frac{35}{5}$

$y = 7$

So Sasha exercises 7 hours per week.

Max:

$x + y = 20$

$x + 7 = 20$

$x = 13$

Max exercises 13 hours a week.

6 0

3 years ago

You will need to flip the inequality sign when you subtract 4 from each side in the inequality y/-2 + 4 < 9.

Talja [164]

Answer:

-1/2y+4<9

-1/2y+4-4<9-4

-1/2<5

(2/-1)*(-1/2y)<(2/-1)*(5)

y>-10

answer is y>-10

Step-by-step explanation:

5 0

3 years ago

A tool store owner bought 18 circular saws for $68 each and 22 cordless drills for $74 each. How much did the owner spend in all

Ivahew [28]

C. They spent $2,852

8 0

3 years ago

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