Question

Identify the vertex, focus, and directrix of the parabola with the equation x^2 – 6x – 8y + 49 = 0

Answer 1

@TuringTest can you help me please??

Answer 2

Have you tried putting this equation into the vertex form?

Answer 3

how do you do that?

Answer 4

@AccessDenied ?

Answer 5

You know the vertex form looks like this?
\( y = a(x - h)^2 + k \)
just to make sure you are familiar with what I am referring to.

Answer 6

yes

Answer 7

So, we could complete the square here:
\( \color{Green}{x^2 - 6x} - 8y + 49 = 0 \)
Which will give us that \( (x - h)^2\) part. Then rearrange so that the y term is alone on the other side of the equation.

Answer 8

ok..then what?

Answer 9

If you do that part, you will obtain the vertex form: \(y = a (x- h)^2 + k\)
From this form, the vertex is directly given by (h, k). And a = 1/(4c), where c is the distance to either focus or directrix.

Answer 10

You use the vertex as the basis point. The distance from vertex to focus is that value of c, and we are moving a vertical distance because the parabola is opening upwards as the focus lies on the axis of symmetry (vertical line through vertex).

Does that make sense? Do I need to cover anything more in-depth? :)