algebra helpp?

Question

does anyone know how to write the equation of a parabola if you're only given the focus and the directrix?

Answer 1

The focus of a parabola is (h, k + p).
The directrix is y = k - p.
Find the vertex (h,k)
The equation of a parabola is (x - h)^2 = 4p(y-k)

AN EXAMPLE:
Focus: (2,5)
Directrix: y = -3
The x-coordinate of the vertex must be 2, because they are both equal to h.
The y-coordinate of the vertex is the average of the y-coordinate of the focus and the directrix.
For V: y = (5 + (-3)) / 2 = 1
V(2,1)
h = 2
k = 1
p = 4

Now, you can plug this into your parabola formula.
(x - h)^2 = 4p(y - k)
(x - (2))^2 = 4(4)(y - 1)
(x - 2)^2 = 16(y - 1)
((x - 2)^2) / 16) = y - 1
((x - 2)^2) / 16) + 1 = y

Hope I helped!
10 points best answer?

Answer 2

Yes, the equation is
x^2=4py, where p is the focus, if focus is (0,p); dir: y=-p
x^2=-4py, if the focus is (0,-p), dir: y=p
y^2=4px, if focus is(p,0), dir: x=-p
y^2=-4px, if focus is (-p,0), dir: x=p

Remember that focus and directrix are working together: if focus is (0,p), then the directrix is y=-p. If you see the same outcome, your conic sections is not shifted and the vertex of a parabola is at the origin. Look at the example of shifted conic sections:

Focus: (3,4) and directrix y=0. By directrix, you know that the function will be looking up or down (I explained why at the top).... As far as the focus' y-component is 4 (focus = 3,4), and is greater that the component of the function of directrix which is zero (y=0), we can say that parabola opens upwards. x^2=4py. Let's now find the shift of the function: Looking both at focus and directrix, y-shift will be: 4-shift = 0+shift, so shift will be equal to 2. So, our p is 2 (which is the y-component of focus minus the shift). Normal x-component of the parabola in our case is zero, so the shift is 3 according to the focus parameters. Therefore, the parabola is x^2=4py ---> (x-3)^2=4*2*(y-2)
(x-3)^2 = 8(y-2). Remember, if shift is 3 units to the right, you subtract 3 from x --> kind of oposite-way: shifting to the positive side will make you subtract, and shifting to the negative side makes you add to the variable.

I hope I am clear enough. if you have any questions, e-mail me...

Eugene.

Answer 3

the equations of a parabola r

1) Upward opening : x ^ 2 = 4*a*y
2) downward : x ^ = -4ay
3)Right handed : y ^ 2 = 4*a*x
4) Left handed : y ^ 2 = - 4*a*x
Thedirectrices r:
1) y = -a
2) y = a
3) x = -a
4) x = a
From the equation u see that a is known equation is known. a is got from focii and directrix. Thus equation can be obtained.

Answer 4

Parabola y^2=4ax