Find an equation of the parabola described and state the two points that define the latus rectum
Focus at (0,4) ; Directrix the line y=4
sorry - the focus cannot be on the directrix
maybe you should fix that.
However, the parabola
(x-h)^2 = 4p(y-k)
has a latus rectum of length 4p.
To find the equation of a parabola when given the focus and directrix, we can use the standard form of a parabolic equation:
(x - h)² = 4p(y - k)
Where (h, k) is the vertex of the parabola, and p is the distance from the vertex to the focus (or from the vertex to the directrix if p is negative).
In this case, the focus is at (0,4), so the vertex is also at (0,4) since the vertex always lies halfway between the focus and the directrix.
The directrix is the line y = 4. Since the directrix is a horizontal line, it means the parabola opens upwards or downwards. In this case, the parabola opens upwards because the vertex has a y-coordinate greater than the directrix.
The distance from the vertex to the directrix is also the distance from the vertex to the focus. In this case, the distance is 4 units, so p = 4.
Now we can write the equation of the parabola:
(x - 0)² = 4(4)(y - 4)
Simplifying the equation:
x² = 16(y - 4)
So the equation of the parabola is x² = 16(y - 4).
To find the two points that define the latus rectum, we can use the formula:
4p
Since p = 4 in this case, the length of the latus rectum is 4p = 16.
Since the parabola opens upwards, we can use the formula for the two points on the parabola that are at a distance of 4p (16 in this case) on each side of the vertex:
(x, y) = (0, 4 ± 8)
So the two points that define the latus rectum are:
(0, 4 + 8) = (0, 12)
(0, 4 - 8) = (0, -4)
Therefore, the two points that define the latus rectum are (0, 12) and (0, -4).