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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 $3x+4y+11=0$?

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  • $\begingroup$ Why the axis has to be horizontal? $\endgroup$ Oct 7, 2019 at 16:47
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    $\begingroup$ "... as the Axis is perpendicular to the directrix ..." This is the correct approach. $\endgroup$
    – Blue
    Oct 7, 2019 at 16:49

1 Answer 1

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The correct answer is $$3x+4y+11=0$$

Note that the axis and the directrix are perpendicular and the focus is on the axis.

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