Find an equation of a parabola with a vertex at the origin and directrix y = –1.5.
A. x = – (1/6)y2
B. x = 1/6y2
C. y = –1/6x2
D. y = 1/6x2
D. y = 1/6x^2
To find the equation of a parabola with a vertex at the origin and a given directrix, we can use the standard form of a parabolic equation:
(y - k)^2 = 4p(x - h)
where (h, k) represents the vertex, and p represents the distance between the vertex and the directrix.
In this case, the vertex is at the origin, so (h, k) = (0, 0). And the directrix is y = -1.5, which means the distance between the vertex and the directrix is the absolute value of -1.5, which is 1.5.
Now we can substitute the values into the equation:
(y - 0)^2 = 4(1.5)(x - 0)
Simplifying further:
y^2 = 6(x - 0)
y^2 = 6x
The equation of the parabola with a vertex at the origin and a directrix y = -1.5 is therefore represented by C. y = -1/6x^2.
To find the equation of a parabola with a vertex at the origin and a directrix at y = -1.5, we can use the standard form of a parabola equation: (y - k)^2 = 4p(x - h), where (h, k) represents the vertex coordinates.
In this case, the vertex is at the origin, so (h, k) = (0, 0).
The directrix is given as y = -1.5, meaning that the distance from any point on the parabola to the directrix is equal to the distance from that same point to the vertex. Since the directrix is below the vertex, the value of p is positive.
Therefore, p = 1.5.
Substituting these values into the standard equation, we get:
(y - 0)^2 = 4(1.5)(x - 0)
which simplifies to:
y^2 = 6x.
So, the equation of the parabola with a vertex at the origin and a directrix at y = -1.5 is:
D. y = 1/6x^2