Question

The equation of a parabola is 24y=(x-2)^2-48. what is the vertex, focus and directrix of the parabola

Answer 1

AAAaannndd the bot gets it wrong yet again!

24y=(x-2)^2-48
(x-2)^2 = 24(y+2)
So we have a vertical axis and p=6, giving us
vertex = (2,-2)
focus = (2,-2+6) = (2,4)
directrix: y = -2-6 = -8

Answer 2

To find the vertex of the parabola, we can rewrite the equation in the standard form:

y = (1/24)(x-2)^2 - 2

The vertex of the parabola is then (2, -2).

To find the focus, we need to first determine whether the parabola opens up or down. Since the coefficient of x^2 is positive, the parabola opens upwards.

The formula for finding the focus of a parabola in standard form is:

F = (h, k + 1/4a)

Where (h, k) is the vertex and a is the coefficient of x^2.

Plugging in the values for our parabola, we get:

F = (2, -2 + 1/(4*1/24))

Simplifying, we get:

F = (2, -1)

So the focus of the parabola is located at (2, -1).

To find the directrix, we need to use the formula:

y = k - 1/4a

Plugging in the values for our parabola, we get:

y = -2 - 1/(4*1/24)

Simplifying, we get:

y = -3

So the directrix of the parabola is the horizontal line y = -3.

Answer 3

I'm sorry for the mistake in my previous response and thank you for providing the correct solution. Yes, you are correct that the vertex is (2,-2), the focus is (2,4), and the directrix is y = -8.