# Find the eccentricity, centre, foci, vertices of the following ellipses and draw the diagram: $9x^{2}+4y^{2}=36$

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}$$=1 • http://clay6.com/mpaimg/15_2_Toolbox.png • Foci(\pm ae,o), center (0,0),vertices (\pm a,0) • End points of Latus Rectum \; (ae,\pm \large\frac{b^2}{a}) and (-ae,\pm \large\frac{b^2}{a}) • Directrices x=\pm \large\frac{a}{e}. • The major axis is y=0 (x- axis) and the minor axes is x=0 (y- axis) • \large\frac{x^2}{b^2}+\frac{y^2}{a^2}$$=1$
• http://clay6.com/mpaimg/15_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:
$9x^2+4y^2=36$
The above equation is divided by 36 we get
$\large\frac{x^2}{4}+\frac{y^2}{9}=$$1 a^2=9,b^2=4 \Rightarrow a=3,b=2 The major axis is the x-axis. Step 2: eccentricity e=\sqrt{1-\large\frac{b^2}{a^2}} \Rightarrow \sqrt{1-\large\frac{4}{9}}=\large\frac{\sqrt 5}{3} Step 3: Centre : Origin (0,0) Step 4: Foci : (0,\pm ae) ae=3\times \large\frac{\sqrt 5}{3}$$=\sqrt 5$
$\Rightarrow (0,\pm \sqrt 5)$
Step 5:
Vertices $(0,\pm a)$
$\Rightarrow (0,\pm 3)$