Find Equation of Parabola Given Focus & Directrix - Jose's Q&A

In summary, the equation of the parabola with focus (3,5) and directrix y=1 is y=\frac{x^2-6x+33}{8}.
  • #1
MarkFL
Gold Member
MHB
13,288
12
Here is the question:

Writing an equation given the directrix and focus?


focus: (3,5) directrix y=1. write an equation for the parabola. How do I do this? please help!

I have posted a link there to this thread so the OP can view my work.
 
Mathematics news on Phys.org
  • #2
Re: jose's question at Yahoo! Questions: find the equation of the parabola given the focus and direc

Hello jose,

Let's let $(x,y)$ be an arbitrary point on the parabola. Now, we know the perpendicular distance from the point to the directrix will be equal to the distance between this point and the focus. Thus, we may state:

\(\displaystyle |y-1|=\sqrt{(x-3)^2+(y-5)^2}\)

Square both sides:

\(\displaystyle (y-1)^2=(x-3)^2+(y-5)^2\)

Expand binomials:

\(\displaystyle y^2-2y+1=x^2-6x+9+y^2-10y+25\)

Collect like terms:

\(\displaystyle 8y=x^2-6x+33\)

Divide through by $8$:

\(\displaystyle y=\frac{x^2-6x+33}{8}\)
 

FAQ: Find Equation of Parabola Given Focus & Directrix - Jose's Q&A

1. What is a focus and directrix in a parabola?

A focus is a fixed point on a parabola that is equidistant from all points on the parabola. The directrix is a fixed line that is perpendicular to the axis of symmetry of the parabola and is equidistant from all points on the parabola.

2. How do you find the equation of a parabola given the focus and directrix?

To find the equation of a parabola given the focus and directrix, you can use the formula:
(y-k)^2 = 4p(x-h), where (h,k) is the coordinates of the vertex, and p is the distance from the vertex to the focus or directrix. If p is positive, the parabola opens upwards, and if p is negative, the parabola opens downwards.

3. Can a parabola have more than one focus and directrix?

No, a parabola can only have one focus and one directrix. These are the defining characteristics of a parabola and cannot be changed.

4. What information is needed to find the equation of a parabola given the focus and directrix?

To find the equation of a parabola given the focus and directrix, you need the coordinates of the focus, the equation of the directrix, and the direction in which the parabola opens (upwards or downwards).

5. How is the focus and directrix related to the shape of a parabola?

The focus and directrix determine the shape and orientation of a parabola. The distance between the focus and directrix is equal, and this distance is also equal to the distance from any point on the parabola to the focus or directrix. The focus and directrix are also used to find the vertex, which is the point where the parabola changes direction.

Similar threads

Replies
2
Views
5K
Replies
6
Views
2K
Replies
5
Views
13K
Replies
2
Views
7K
Replies
1
Views
7K
Replies
1
Views
4K
Replies
4
Views
2K
Back
Top