Question

Find the coordinates of focus, the equation of the directrix and length of the latus rectum of the conic represented by the equation $3x^2=-y$

Answer 1

The given equation is $x^2=-\frac{1}{3}y$ ..........$\left(1\right)$

which is comparable with $x^2=-4ay$, so $\left(1\right)$ represents a parabola of the standard form.

Here $4a=\frac{1}{3}\Rightarrow a=\frac{1}{12}$

$\therefore$The focus is $(0,-a)=(0,-\frac{1}{12})$ and the equation of the directrix is $y-a=0$

$\Rightarrow y-\frac{1}{12}=0$ or $12y-1=0$

The length of the latus rectum$=4a=4\times \frac{1}{12}=\frac{1}{3}$