Find the equations of directrices, latus rectum and lengths of latus rectums of the following ellipses: $9x^{2}+16y^{2}=144$

Toolbox:
• Standard forms of equation of the ellipse with major axes 2a, minor axis 2b $(a >b)$ eccentricity e and $b^2=a^2(1-e^2)$ or $e^2=1-\large\frac{b^2}{a^2}$
• $\large\frac{x^2}{a^2}+\frac{y^2}{b^2}$$=1 • http://clay6.com/mpaimg/10_2_Toolbox.png • Foci(\pm ae,o), center (0,0),vertices (\pm a,0) • End points of Latus Rectum \; (ae,\pm \large\frac{b^2}{a}) and (-ae,\pm \large\frac{b^2}{a}) • Directrices x=\pm \large\frac{a}{e}. • The major axis is y=0 (x- axis) and the minor axes is x=0 (y- axis) • \large\frac{x^2}{b^2}+\frac{y^2}{a^2}$$=1$
• http://clay6.com/mpaimg/10_2_Toolbox1.png
• Foci$(0,\pm ae),$ center $(0,0)$,vertices $(0,\pm a)$
• End points of Latus Rectum $(\pm \large\frac{b^2}{a}$$,ae) and (\pm \large\frac{b^2}{a}$$,-ae)$
• Directrices $y=\pm \large\frac{a}{e}$.
• The major axis is $x=0$ (y- axis) and the minor axes is $y=0$ (x- axis)
Step 1:
$\large\frac{x^2}{16}+\frac{y^2}{9}$$=1 a^2=16,b^2=9 \Rightarrow a=4,b=3 The major axis is the x-axis,y=0 Step 2: Eccentricity e=\sqrt{1=\large\frac{b^2}{a^2}} \Rightarrow \sqrt{1-\large\frac{9}{16}}=\large\frac{\sqrt 7}{4} Step 3: Directrices x=\pm\large\frac{a}{e} x=\pm \large\frac{4}{\Large\frac{\sqrt 7}{4}}\Rightarrow \pm\large\frac{16}{\sqrt 7} Step 4: Latus rectum x=\pm ae x=\pm 4.\large\frac{\sqrt 7}{4}$$\Rightarrow x=\pm \sqrt 7$
Step 5:
Length of $LR$ =$\large\frac{2b^2}{a}$
$\Rightarrow \large\frac{2\times 9}{4}=\large\frac{9}{2}$