A parabola has a directrix y=-3, axis =4 and latus rectum equal to 2. find its equation.
Please help me its for my homework
1/a = 2.
a = 1/2.
1/4a = 1/2.
k- (-3) = 1/4a = 1/2.
k + 3 = 1/2
k = 1/2 - 6/2 = -5/2.
h = axis = 4.
Y = Y = a(x-h)^2 + k.
Y = (1/2)(x-4)^2 - 5/2.
To find the equation of the parabola, we'll first need to understand the properties of a parabola.
A parabola is defined by its vertex and its directrix. The vertex is a point on the parabola, and the directrix is a line that is equidistant from all points on the parabola. In this case, we are given the directrix as y = -3.
The axis of the parabola is a vertical line passing through the vertex. It divides the parabola into two symmetric halves. In this case, we are given the axis as x = 4.
The latus rectum is a line segment perpendicular to the axis, passing through the focus and having its endpoints on the parabola. It is twice the focal length of the parabola. In this case, we are given the latus rectum as 2 units.
To find the equation of the parabola, we need to determine the coordinates of the vertex and the focal point.
First, let's find the vertex. Since the axis is given as x = 4, the x-coordinate of the vertex is 4. To find the y-coordinate, we need to determine the distance between the axis and the directrix, which is the same as the distance between the vertex and the directrix.
Since the directrix is a horizontal line, the distance between the vertex and the directrix is the absolute value of the difference between the y-coordinate of the directrix and the y-coordinate of the vertex.
The y-coordinate of the directrix is -3, and the y-coordinate of the vertex is not given. So, we'll let it be some variable, say y_v.
The distance between the vertex (x_v, y_v) and the directrix y = -3 is:
|y_v - (-3)| = |y_v + 3|
Since the vertex is equidistant from the directrix and the focus, we can also find the distance between the vertex and the focus using the given latus rectum.
The distance between the vertex (x_v, y_v) and the focus (x_f, y_f) is 1/2 latus rectum = 1/2 * 2 = 1 unit.
Using the distance formula, the square of the distance between the vertex and the focus is given by:
(x_f - x_v)^2 + (y_f - y_v)^2 = 1
Substituting the coordinates of the vertex (4, y_v) and the coordinates of the focus (x_f, y_f), the equation becomes:
(x_f - 4)^2 + (y_f - y_v)^2 = 1
Now, we have two equations:
1) |y_v + 3| = 1/2 latus rectum = 1
2) (x_f - 4)^2 + (y_f - y_v)^2 = 1
Solving these equations will give us the coordinates of the vertex and the focus, which we can then use to write the equation of the parabola.
Since you have provided the latus rectum as 2, let's substitute that value into the equations and solve them.
To find the equation of a parabola given its directrix, axis, and latus rectum, we can use the standard equation of a vertical parabola:
(x - h)^2 = 4p(y - k)
where (h, k) represents the vertex, and p is the distance from the vertex to the focus and the vertex to the directrix.
In this case, we know that the directrix is y = -3 and the axis is x = 4, meaning the vertex lies on the line x = 4.
Since the latus rectum is given as 2, we know that the distance from the vertex to the focus (p) is 1.
Therefore, the vertex will be the point (4, k), and the focus will be (4, k + p).
Let's first find the value of k:
Since the directrix is y = -3, we can say that the equation of the directrix is y + 3 = 0.
Since the vertex lies on the line x = 4, the coordinates of the vertex will be (4, k), meaning that the parabola at the vertex should follow the equation (x - 4)^2 = -4p(y - k).
Substituting the values of the vertex (4, k), we get:
(4 - 4)^2 = -4p(k + p - k)
0 = -4p
Therefore, p must be 0 for the equation of the directrix to hold.
Now let's substitute the vertex (4, k) and the focus (4, k + p) into the general equation to find the equation of the parabola:
(x - 4)^2 = 4(0)(y - k)
(x - 4)^2 = 0
Thus, the equation of the given parabola is (x - 4)^2 = 0.