Question

How do you write an equation of a parabola with a vertex at the origin and a directrix at \(y = 5\)?

a) \(y = x^2 + 5\)\
b) \(y = -x^2 + 5\)\
c) \(y = x^2 - 5\)\
d) \(y = -x^2 - 5\)

Answer 1

To write the equation of a parabola with a vertex at the origin and a directrix at y = 5, we can use the standard form of the equation and the formula for the distance from a point to a line. The equation of the parabola is y = (1/5)x².

Step-by-step explanation:

To write the equation of a parabola with a vertex at the origin and a directrix at y = 5, we can start by considering the standard form of the equation of a parabola, which is y = ax².

Since the vertex is at the origin, the equation becomes y = ax². Next, we can use the formula for the distance from a point to a line to find the value of a.

The distance from the vertex (0,0) to the directrix y = 5 is |0 - 5| = 5. Substituting this into the formula, we get a(0 - 5)² = 5, which simplifies to 25a = 5. Solving for a, we find that a = 1/5. Therefore, the equation of the parabola is y = (1/5)x².