I have to write the equation of teh parabola with the fiven focus and directrix.

Question

i have to write the equation of the parabola with the given focus, (0,2), and the directrix, y= -2. I also have to graph the equation, but in order to do that don't i atleast need two points of the parabola?

Answer 1

Given parabola focus (0,2) and directrix y = -2.

Parabola equation x^2= 4ay, focus (0,a) when a>0 then x = 0, y = -a

Substitute the a value in parabola equation.

Parabola equation is x^2 = 8y

To find orderd pairs x = 0,4,-4 substitute respectively.

For x = 0 ⇒ 0 = 8y

y = 0

For x = 4 ⇒ 16 = 8y

y = 16/2 = 8

y = 2

For x = -4 ⇒ 16 = 8y

y = 16/8 = 2

y = 2

We can able to write orderd pairs (0,0) , (4,2), (-4,2).

Graph of parabola

Draw the coordinate plane.

Plot the ponits and connect it.

Observe the graph focus and directrix of parabola.

Answer 2

Information About Parabolas:

Form of Equation	(x - h)^2 = 4p (y - k)	(y - k)^2 = 4p (x - h)
Vertex	(h, k)	(h, k)
Axis of Symmetry	x = h	y = k
Focus	(h, k + p)	(h + p, k)
Directrix	y = k - p	x = h - p
Direction of Opening	upward if p > 0, downward if p < 0	right if p > 0, left if p < 0
Length of Latus Rectum	| 4p | units	| 4p | units

The focus of the parabola is (0, 2) and directrix y = - 2.

Since the directrix is y = -2, the standard form of parabola equation is (x - h)^2 = 4p (y - k).

Find the value of h, k and p as follows :

The equation of directrix  y = k - p = - 2 ⇒ k - p = - 2 ------> (1).

Focus = (h, k + p) = (0, 2) ⇒ h = 0 and k + p = 2       ------> (2).

Solve equation 1 and 2, to obtain k = 0 and p = 2.

The standard form of parabola equation is (x - 0)^2 = 4(2) (y - 0) ⇒ x² = 8y.

Vertex = (h, k) = (0, 0) and the axis of symmetry is x = h = 0.

Graph :

Draw a coordinate plane, and plot the vertex (0, 0) and draw the the axis of symmetry x = 0 in the plane.

The axis of symmetry divides the parabola into two equal parts. So if there is point on one side, there is a corresponding point on the other side that is the same distance from the axis of symmetry and has the same y - value.

Let' s find another point. Choose an x - value of 1 and substitute in the parabola equation. x² = 8y ⇒ (1)² = 8y ⇒ y = 1/8. The new point is (1, 1/8), the distance between point (1, 1/8) and the axis of symmetry is 1 unit. The point paired with it on the other side of the axis of symmetry is (-1, 1/8).

Repeat this and choose an x - value of 2 and substitute in the parabola equation. x² = 8y ⇒ (2)² = 8y ⇒ y = 1/2. The new point is (2, 1/2), the distance between point (2, 1/2) and the axis of symmetry is 2 units. The point paired with it on the other side of the axis of symmetry is (-2, 1/2).

Connect these points and create a smooth curve.