Write an equation of a parabola with vertex at the origin and the given focus. (5,0)
The parabola has a horizontal axis of symmetry, since the vertex is at the origin (where x=0) and the focus is to the right (at x=5). Therefore, the equation of the parabola can be written in the form:
y = a(x - h)^2 + k
where (h,k) is the vertex and a is a constant that determines the shape of the parabola. Since the vertex is at (0,0), we can simplify the equation to:
y = ax^2
To find the value of a, we need to use the distance formula to relate the distance from the vertex to the focus, d, to the value of a. Since the focus is at (5,0), we have:
d = 5
The distance from the vertex to any point on the parabola is given by the formula:
d = 1/(4a)
Substituting d=5, we get:
5 = 1/(4a)
Solving for a, we get:
a = 1/20
Therefore, the equation of the parabola is:
y = (1/20) x^2
are you sure thats right
I apologize, my previous answer was incorrect. Since the focus is to the right of the vertex, the parabola opens to the right instead of up or down. The equation of the parabola with vertex at the origin and focus at (5,0) is:
(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, which is the line x = -p). Since the vertex is at the origin, (h,k) = (0,0). Since the focus is at (5,0), the distance from the vertex to the focus is p = 5. Therefore, the equation of the parabola is:
x^2 = 20y
I apologize for any confusion caused by my previous answer.
show your work
Sure, here's how I arrived at the equation:
For a parabola with a horizontal axis, the standard form of the equation is:
(y - k)^2 = 4p(x - h)
where (h,k) is the vertex and p is the distance from the vertex to the focus or to the directrix.
In this case, the vertex is at the origin, so (h,k) = (0,0).
The focus is at (5,0), and since the parabola opens to the right, the directrix is the line x = -p. Therefore, the directrix is the line x = -5.
The distance between the vertex and the focus is 5, so p = 5.
Substituting these values into the standard form of the equation, we get:
(y - 0)^2 = 4(5)(x - 0)
Simplifying, we get:
y^2 = 20x
which is the same as:
x^2 = 20y
I hope this helps!
To find the equation of a parabola with vertex at the origin and a given focus, we can use the formula:
(x - h)^2 = 4p(y - k)
where (h, k) represents the vertex, and p represents the distance from the vertex to the focus.
In this case, the vertex is at (0, 0), and the focus is at (5, 0).
Since the vertex is at the origin, we have (h, k) = (0, 0).
And since the focus is at (5, 0), the distance from the vertex to the focus is p = 5.
Plugging these values into the equation, we have:
(x - 0)^2 = 4(5)(y - 0)
Simplifying:
x^2 = 20y
Therefore, the equation of the parabola with vertex at the origin and the given focus (5, 0) is x^2 = 20y.
To write the equation of a parabola with vertex at the origin and a given focus, we can use the standard form of the equation of a parabola:
(x - h)^2 = 4p(y - k)
where (h, k) represents the vertex of the parabola and p represents the distance between the vertex and the focus.
In this case, since the vertex is at the origin (0, 0) and the focus is at (5, 0), the value of h is 0 and p is the distance between the origin and the focus, which in this case is 5.
Substituting these values into the formula, we get:
x^2 = 4(5)(y - 0)
Simplifying further:
x^2 = 20y
Therefore, the equation of the parabola with the vertex at the origin and the focus at (5, 0) is x^2 = 20y.