Derive the equation of the parabola with a focus at (0, 1) and a directrix of y = -1. ​

The equation of the parabola is y = x²/4

Step-by-step explanation:

The given focus of the parabola = (0, 1)

The directrix of the parabola is y = -1

A form of the equation of a parabola is presented as follows;

(x - h)² = 4·p·(y - k)

We note that the equation of the directrix is y = k - p

The focus = (h, k + p)

Therefore, by comparison, we have;

k + p = 1...(1)

k - p = -1...(2)

h = 0...(3)

Adding equation (1) to equation (2) gives;

On the left hand side of the addition, we have;

k + p + (k - p) = k + k + p - p = 2·k

On the right hand side of the addition, we have;

1 + -1 = 0

Equating both sides, gives;

2·k = 0

∴ k = 0/2 = 0

From equation (1)

k + p = 0 + 1 = 1

∴ p = 1

Plugging in the values of the variables, 'h', 'k', and 'p' into the equation of the parabola, (x - h)² = 4·p·(y - k), gives;

(x - 0)² = 4 × 1 × (y - 0)

∴ x² = 4·y

The general form of the equation of the parabola, y = a·x² + b·x + c, is therefore;

y = x²/4.

9.55 degrees.

step-by-step explanation:

the given angle is

1 cycle is equal to 2π radians. so 4π will be equal to two cycles.

5.166667π-4π = 1.166667π

hence, this angle goes a little more than negative x axis.

1.166667π-π= 0.166667π

hence, the reference angle is 0.166667π radians.

1 radian = 57.2958 degrees

now, in degrees, this should be equals to 9.55 degrees.

x = ±sqrt(10)

step-by-step explanation:

5 - 2x² = -15

subtract 5 from each side

5-5 - 2x² = -15-5

-2x^2   = -20

divide by -2 on each side

-2x^2   /-2= -20/-2

x^2 =10

take the square root on each side

sqrt(x^2 )=±sqrt(10)

x = ±sqrt(10)

check:

5 - 2x² = -15

+sqrt(10)

5 - 2(sqrt(10))² = -15

5 -2(10) = -15

5 -20 = -15

-15 = -15

-sqrt(10)

5 - 2(-sqrt(10))² = -15

5 -2(10) = -15

5 -20 = -15

-15 = -15