Question

Write the equation for a parabola with a focus at (-2, 5) and a directrix at y = 3. MUST BE IN FORMAT: y = (insert here)

A. y = 1/4(x + 2)^2 + 3
B. y = -1/4(x + 2)^2 + 5
C. y = 1/4(x + 2)^2 - 3
D. y = -1/4(x + 2)^2 - 5

Answer 1

The equation for the parabola with a focus at (-2, 5) and directrix at y = 3 is y = (1/4)(x + 2)^2 + 3, which corresponds to answer option A.

Step-by-step explanation:

To find the equation of a parabola with a focus at (-2, 5), and a directrix at y = 3, we start by noting the vertex of the parabola is midway between the focus and directrix. Hence, the vertex is at (-2, 4). Since the parabola opens upwards (focus above directrix), the general form of the equation is:

y - k = a(x - h)^2, where (h, k) is the vertex.

Here, (h, k) = (-2, 4) and the distance between the focus and directrix, which is also 2 times the value of 'a', is 5 - 3 = 2. Therefore, 'a' is 1/2. Plugging in the values we get:

y - 4 = ​​(1/4)(x + 2)^2

Finally, solving for y yields:

y = ​​(1/4)(x + 2)^2 + 4

Since the answer needs to match one of the options given, which are in terms of y = a(x + 2)^2 ​​+ b, we rewrite it as:

y = ​​(1/4)(x + 2)^2 + 3

This corresponds to answer option A.