Find the coordinates of the focus and equation of the directrix for the parabola given by
y2 = −4x.

the general formula for this parabola is y2 = 4px.

therefore, the value of p is

the coordinates of the focus are

the equation of the directrix is

The answers to your three problem question are shown

1) the value of p is

p = -1

2) The coordinates of the focus are

Focus =  (-1,0)

3) The equation of the directrix is

Directrix

x = 1

Step-by-step explanation:

The general equation for this parabola is

y^2 = 4px

Problem 1

Find the value of p.

We are told that the equation of the problem is

y^2 = -4x

And the general formula is

y^2 = 4px

From there, we can deduce that

p = -1, because

y^2 = 4(-1)x = -4x

This means that p = -1

Problem 2

Find the focus

To find the focus we can see the equations attached below for the focus, vertex and directrix.

In these case, the equations still apply, even though the variable is inverted, we just need to adjust it

y^2 = -4x =>

x = (-1/4)*y^2

x = a*y^2 + b*y +c

Focus

((4ac -b^2  + 1)/4a, -b/2a)

But, b = 0 and c = 0

=>

(1/4a,0) = (1/4(-1/4),0) = (-1,0)

Focus =  (-1,0)

Problem 3

Find the directrix

The equation is

x = c - (b^2 + 1).4a

But, b = 0 and c = 0

x = -4*a

x = -4* (-1/4) = 1

x = 1

Directrix

x = 1

a

step-by-step explanation:

a

{x = -1/2 ,   y = -1

step-by-step explanation:

solve the following system:

{2 x + y = -2

{10 x - 3 y = -2

in the first equation, look to solve for y:

{2 x + y = -2

{10 x - 3 y = -2

subtract 2 x from both sides:

{y = -2 x - 2

{10 x - 3 y = -2

substitute y = -2 x - 2 into the second equation:

{y = -2 x - 2

{10 x - 3 (-2 x - 2) = -2

10 x - 3 (-2 x - 2) = (6 x + 6) + 10 x = 16 x + 6:

{y = -2 x - 2

{16 x + 6 = -2

in the second equation, look to solve for x:

{y = -2 x - 2

{16 x + 6 = -2

subtract 6 from both sides:

{y = -2 x - 2

{16 x = -8

divide both sides by 16:

{y = -2 x - 2

{x = -1/2

substitute x = -1/2 into the first equation:

{y = -1

{x = -1/2

collect results in alphabetical order:

answer:   {x = -1/2 ,   y = -1