How to identify properties of conic $12x+y^2-6y+45=0$

Question

I need to find out the type of conic, the coordinates of the center, focus (foci), vertex (vertices), directrix for the conic given by:

$$12x+y^2-6y+45=0$$

I completed the square to get

$$(y-3)^2+36=-12x$$

This is a parabola I believe since it is of the form $y^2=4px$ which is a parabola rotated by $\frac{\pi}{2}$ clockwise if I'm not mistaken.

I am now struggling on how to get the rest of this information so any help is appreciated.

Answer 1

Hints and References :

• Look up the detailed guide here.
• The parabola is shifted and it is rotated by 180 degrees . ( The -ve sign of p indicates that it has been flipped)
• This might also help.

Answer 2

Yes, this is a parabola. You can factor out like this:

$$(y-3)^2 = -12(x+3)$$

This means, the vertex is located at $(-3,3)$. The parabola is opening towards the left.

In this case, $-4p = -12$, $p=3$. This means the distance from focus to vertex is 3. If you draw the parabola, the focus should be located at $(-6,3)$. The directrix should be a vertical line with the nearest point to the vertex also having a distance of 3. Hence, directrix is $x=0$.