Question:
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola :
$5 y^{2}=-16 x$
Solution:
Given equation :
$5 y^{2}=-16 x$
$y^{2}=-\frac{16}{5} x$
Comparing the given equation with parabola having an equation,
$y^{2}=-4 a x$
$\cdot 4 a=\frac{16}{5}$
$\cdot a=\frac{4}{5}$
Focus: $F(-a, 0)$
$=F\left(-\frac{4}{5}, 0\right)$
Vertex :
$\mathrm{A}(0,0)=\mathrm{A}(0,0)$
Equation of the directrix :
$x-a=0$
$x-\frac{4}{5}=0$
$x=\frac{4}{5}$
Lenth of latusrectum : $4 a=\frac{16}{5}$
