Find the equation of a parabola with vertex (3,1) and focus (5,1)
note that the tw points lie on the same horizontal line. The parabola
y^2 = 4px has
vertex at (0,0)
focus at (p,0)
distance from focus to vertex = p.
So, shift this to (3,1) and p = 2
and you have
(y-1)^2 = 8(x-3)
See properties at
https://www.wolframalpha.com/input/?i=parabola+%28y-1%29%5E2+%3D+8%28x-3%29
Oh, a parabola with a vertex and a focus, huh? Sounds like it needs a little bit of mathematical clowning around. Don't worry, I've got this!
Now, the general equation of a parabola is of the form y = a(x-h)^2 + k, where (h, k) represents the vertex.
So we know the vertex is at (3, 1). That means our equation starts off like y = a(x - 3)^2 + 1.
Now, let's deal with the focus point, which is (5, 1). The focus point is always a distance of 1/a from the vertex. So in this case, the focus point is 2 units away from the vertex. That means, 2 = 1/a.
Hold on, I need to put on my clown thinking cap. *Honk honk* Okay, that means a = 1/2.
Plugging it into our equation, we get y = (1/2)(x - 3)^2 + 1.
And voila! There you have it, the equation of the parabola with the given vertex and focus is y = (1/2)(x - 3)^2 + 1. Now you've got a mathematical clown-approved answer!
To find the equation of a parabola with the given vertex and focus, we can use the standard equation of a parabola:
(x - h)^2 = 4p(y - k)
where (h, k) is the vertex coordinates and p is the distance between the focus and vertex.
In this case, the vertex is (3, 1) and the focus is (5, 1). Since the x-coordinates of both the vertex and focus are the same, we know that the parabola opens to the right.
Now, let's find the value of p (the distance between the focus and vertex):
p = |5 - 3| = 2
Therefore, the equation of the parabola is:
(x - 3)^2 = 4(2)(y - 1)
Simplifying further:
(x - 3)^2 = 8(y - 1)
To find the equation of a parabola given its vertex and focus, we'll use the formula for the equation of a parabola with a vertical axis of symmetry:
(x - h)^2 = 4p(y - k)
Where (h, k) is the vertex and p is the distance from the vertex to the focus (or from the vertex to the directrix).
In this case, the vertex is (3, 1) and the focus is (5, 1). The parabola has a vertical axis of symmetry, so the equation will take the form:
(x - 3)^2 = 4p(y - 1)
Now, we need to find the value of p. To do this, we need to calculate the distance from the vertex to the focus. We can use the distance formula:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Distance = √[(5 - 3)^2 + (1 - 1)^2]
Distance = √[2^2 + 0^2]
Distance = √4
Distance = 2
The distance from the vertex to the focus is 2. Since the parabola opens to the right (based on the vertex and focus), p will be positive. Therefore, p = 2.
Substituting p = 2 into the equation, we get:
(x - 3)^2 = 4(2)(y - 1)
Simplifying:
(x - 3)^2 = 8(y - 1)
So, the equation of the parabola is (x - 3)^2 = 8(y - 1).