Find the vertex and focus of the parabola.
(x^2)-(24y) = 0
X^2 - 24Y = 0.
24Y = X^2,
Y = X^2 / 24,
Y = 1/24 (X^2).
Vertex Form: Y = a(x - h)^2 + k.
h = 0, k = 0. a = 1/24.
V(h , k) = V(0 , 0).
The distance from vertex to focus point = 1/4a.
F(0 , Yf),
Yf - k = 1/4a,
Yf - 0 = 1/4a,
Yf = 1/4a.
4a = 4 * 1/24 = 4/24 = 1/6.
Yf = 1/4A = 6/1 = 6.
F(h , Yf) = F(0 , 6).
Nice one
To find the vertex and focus of the parabola given by the equation (x^2) - 24y = 0, let's start by rearranging the equation in vertex form.
The vertex form of a parabola is given by (x - h)^2 = 4a(y - k), where (h, k) represents the coordinates of the vertex, and "a" is a constant.
Let's manipulate the given equation to this form:
(x^2) = 24y
Divide both sides of the equation by 24 to isolate y:
(x^2) / 24 = y
Now, rewrite the equation by switching the x and y variables:
(y - 0) = (x^2) / 24
Comparing this form with the vertex form, we can see that the vertex is at (h, k) = (0, 0), since it matches the form (x - h)^2 = 4a(y - k).
To find the focus of the parabola, we can use the equation c = a / (4 * h), where "c" is the distance from the vertex to the focus.
Given that a = 1 (the coefficient in front of y) and h = 0 (the x-coordinate of the vertex), we can calculate the value of c:
c = 1 / (4 * 0)
c = 1 / 0 (undefined)
Since "c" is undefined, it means that the parabola does not have a specific focus.
Therefore, the vertex of the parabola is (0, 0), and it does not have a focus.
To find the vertex and focus of a parabola, we need to put the equation in the standard form, which is of the form:
(x - h)^2 = 4p(y - k)
where (h, k) is the vertex, and p is the distance between the vertex and the focus.
Let's rearrange the given equation:
x^2 - 24y = 0
To rewrite it in the standard form, we need to complete the square.
The first step is to move the "x^2" term to the other side of the equation:
x^2 = 24y
Now, let's complete the square by adding a constant to both sides of the equation that will make the left side a perfect square trinomial:
x^2 = 24y + c
To determine the value of c, we need to take half of the coefficient of y (which is 24) and square it. Half of 24 is 12, and 12 squared is 144. So, we add 144 to both sides of the equation:
x^2 = 24y + 144
To make the left side a perfect square, we rewrite it as (x - 0)^2. Since the coefficient of y is 24, we can rewrite the right side divided by 24:
(x - 0)^2 = 24/24 * (y + 144/24)
Simplifying further, we have:
(x - 0)^2 = y + 6
Comparing this with the standard form, we find that:
h = 0 (x-coordinate of the vertex),
k = -6 (y-coordinate of the vertex),
and p = 1/4 (distance from the vertex to the focus).
Therefore, based on the standard form:
Vertex: (h, k) = (0, -6)
Focus: (h, k + p) = (0, -6 + 1/4) = (0, -6.25)
So, the vertex of the parabola is (0, -6), and the focus is located at (0, -6.25).