Directrix at y=3,axis x=2 Latus rectum 4
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To find the focus and vertex, we can start by identifying the equation of the parabola.
The vertex form of a parabola is given by:
(y - k) = a(x - h)^2
Given that the directrix is y = 3 and the latus rectum is 4, we can determine the value of "a" in the equation.
The latus rectum is the line segment passing through the focus, perpendicular to the axis of symmetry, and has a length of 4a. In this case, the length is given as 4, so 4a = 4, and a = 1.
Therefore, the equation of the parabola in vertex form is:
(y - k) = (x - h)^2
We can substitute the values of (h, k) for the vertex, and (x, y) for a point on the parabola to determine the values of h and k.
We know that the axis of symmetry is x = 2, which is also the x-coordinate of the vertex. So, h = 2.
Now, let's find the y-coordinate of the vertex, k.
Since the directrix is y = 3, the distance between the vertex and the directrix is the same as the distance between the vertex and the focus.
The distance formula is given by:
Distance = |y2 - y1| / sqrt(1 + m^2)
In this case, m is the slope of the parabola, which is the derivative of the equation. Since a = 1, the equation becomes:
(y - 0) = (x - 0)^2
Differentiating both sides, we get:
1 = 2x
x = 1/2
This is the x-coordinate of the focus. Now, let's find the y-coordinate of the focus.
Substituting x = 1/2 into the equation of the parabola, we get:
(y - 0) = (1/2 - 0)^2
y = 1/4
The distance between the focus and the directrix is |1/4 - 3| = 11/4.
Since the vertex is equidistant from the focus and the directrix, we can calculate the y-coordinate of the vertex:
k = (1/4 + 3) / 2
k = 13/8
Therefore, the vertex is (2, 13/8) and the focus is (1/2, 1/4)
To find the equation of the parabola given the directrix at y = 3, the axis along x = 2, and the length of the latus rectum (a.k.a. focal width) as 4 units, you can follow these steps:
Step 1: Identify the focus point.
Since the directrix is a horizontal line at y = 3 and the axis is a vertical line at x = 2, the focus point will be (2, k), where k is the distance between the vertex and the focus.
Step 2: Find the distance from the vertex to the focus.
The latus rectum is 4 units in length, and it is defined as the distance between the focus and the directrix. Hence, the distance from the vertex to the focus is half the latus rectum since the parabola is symmetric. Therefore, k = 2 units.
Step 3: Determine the equation of the parabola.
Since the vertex lies on the axis of symmetry, and we know the vertex as (h, k) = (2, 3), the equation of the parabola in vertex form is:
(x-h)^2 = 4a(y-k)
Substituting the given values:
(x-2)^2 = 4(2)(y-3)
Simplifying:
(x-2)^2 = 8(y-3)
So, the equation of the parabola with the given properties is (x-2)^2 = 8(y-3).