Write equation of parabolla focus (5,0) and p:4 vertical axis
sorry, do not know what p:4 means
Nmnb
To find the equation of a parabola with a vertical axis and given focus at (5,0) and p = 4, we first need to determine the value of the parameter a.
The standard equation for a parabola with a vertical axis is given by:
(x - h)² = 4a(y - k)
Where (h,k) represents the coordinates of the vertex.
Since the vertex lies on the axis of symmetry, we know that its x-coordinate will be equal to the x-coordinate of the focus, which is 5. Therefore, h = 5.
Next, we need to find the y-coordinate of the vertex. The vertex lies halfway between the focus and the directrix. Considering the directrix is a vertical line, it will have the equation x = -p = -4.
Since the directrix is 4 units away from the focus and the focus lies on the positive x-axis, the vertex will have a y-coordinate of 4 units below the focus. Hence, k = -4.
Now, we can substitute the values of h, k, and p into the standard equation to obtain the equation of the parabola:
(x - 5)² = 4a(y - (-4))
(x - 5)² = 4a(y + 4)
(x - 5)² = 4ay + 16a
Therefore, the equation of the parabola with focus (5,0) and p = 4 is (x - 5)² = 4ay + 16a.
To write the equation of a parabola given its focus and vertical axis, you can use the standard form of the equation for a parabola:
(x – h)^2 = 4p(y – k)
where (h, k) represents the vertex coordinates, p is the distance from the vertex to the focus/focal length, and the axis of symmetry is parallel to either the x-axis (if it is a vertical parabola) or the y-axis (if it is a horizontal parabola).
In this case, the vertex is not given, but we are given the focus (5, 0) and the value of p, which is 4. Since the parabola has a vertical axis, we can determine that the vertex will have the form (h, k) = (h, 0).
So, using the given values, we have:
(x – h)^2 = 4p(y – k)
(x – h)^2 = 4p(y – 0)
(x – h)^2 = 4p(y)
Substituting the value of p=4, we get:
(x – h)^2 = 4(4)(y)
(x – h)^2 = 16y
Since the focus is at (5, 0), it means that the distance from the vertex to the focus is equal to p, which is 4. So, the x-coordinate of the vertex is h = 5, and the equation becomes:
(x – 5)^2 = 16y
Therefore, the equation of the parabola with focus (5, 0), and p = 4, with a vertical axis, is (x – 5)^2 = 16y.