I want to find the equation of the axis of a parabola with focus $(-1,-2)$ and directrix $4x-3y+2=0$.
What I was thinking is that, as the axis is horizontal, its slope is $0$, and it passes through the focus, and hence I could find its equation in the point-slope form, and got the equation to be $y=-2.$
But, then again, as the axis is perpendicular to the directrix, its equation should be $-3x-4y+k=0$. I found $k= -11$. So, the equation becomes $3x+4y+11=0$.
I can't get which of two equations is correct: $y = -2$ or $3x+4y+11=0$?
The correct answer is $$3x+4y+11=0$$
Note that the axis and the directrix are perpendicular and the focus is on the axis.