# Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum for the following : $x^2=-9y$

$\begin {array} {1 1} (A)\;focus \bigg(\large\frac{9}{4}, 0 \bigg) , Axis : x - axis, Equation \: of \: directrix : 4x-9=0, Length \: of \: the \: latus \: rectum : 9 \\ (B)\; focus \bigg(0, \large\frac{9}{4} \bigg) , Axis : y - axis, Equation \: of \: directrix : 4y+9=0, Length \: of \: the \: latus \: rectum : 9 \\ (C)\;focus \bigg(0, -\large\frac{9}{4} \bigg) , Axis : y - axis, Equation \: of \: directrix : 4y-9=0, Length \: of \: the \: latus \: rectum : 9 \\ (D)\;focus \bigg(-\large\frac{9}{4}, 0) , Axis : x - axis, Equation \: of \: directrix : 4x+9=0, Length \: of \: the \: latus \: rectum : 9 \end {array}$

Toolbox:
• Equation of the parabola $x^2=-4ay$
• Since the coefficient of y is negative the curve is open downwards.
• Coordinates of focus is (0, -a)
• Length of the latus rectum is 4a.
• Equation of directrix is y-a = 0
Step 1 :
The given equation is $x^2=-9y$
The coefficient of y is negative. Hence the parabola opens downwards.
Comparing this equation with
$x^2=-4ay$
$4a=9$
$\Rightarrow a = \large\frac{9}{4}$
Hence the coordinates of focus are
focus : $\bigg( 0, -\large\frac{9}{4} \bigg)$
Since the given equation involves $x^2$ , the axis of the parabola is y - axis.
Axis : y - axis
Equation of the directrix : y = a
(i.e) $y = \large\frac{9}{4}$
or $y -\large\frac{9}{4}$$=0 or 4y-9=0 Length of the latus rectum : 4a 4 \times \large\frac{9}{4}$$=9$