Find the coordinates of the focus,axis of the parabola,the equation of the directrix,and the length of the latus rectum for $y^{2} = -8x$.
(N/A) The given equation is $y^{2} = -8x$.
Here,the coefficient of $x$ is negative,so the parabola opens towards the left.
On comparing this equation with the standard form $y^{2} = -4ax$,we obtain:
$-4a = -8 \Rightarrow a = 2$.
$\therefore$ The coordinates of the focus are $(-a, 0) = (-2, 0)$.
Since the equation involves $y^{2}$,the axis of the parabola is the $x$-axis.
The equation of the directrix is $x = a$,which gives $x = 2$.
The length of the latus rectum is $4a = 4(2) = 8$.
Here,the coefficient of $x$ is negative,so the parabola opens towards the left.
On comparing this equation with the standard form $y^{2} = -4ax$,we obtain:
$-4a = -8 \Rightarrow a = 2$.
$\therefore$ The coordinates of the focus are $(-a, 0) = (-2, 0)$.
Since the equation involves $y^{2}$,the axis of the parabola is the $x$-axis.
The equation of the directrix is $x = a$,which gives $x = 2$.
The length of the latus rectum is $4a = 4(2) = 8$.
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