Q.Find the length of major axis, minor axis, latus rectum, eccentricity, coordinates of centre, foci and the equations of directrices of the ellipse $9x^2 + 16y^2 = 144$.

Convert to standard form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and read off each quantity from $a,b$.

Divide $9x^2+16y^2=144$ by $144$:

$\frac{x^2}{16}+\frac{y^2}{9}=1.$

So $a^2=16\ (a=4)$ and $b^2=9\ (b=3)$, with the major axis along the $x$-axis since $a>b$.

• Major axis length $=2a=8$.
• Minor axis length $=2b=6$.
• Latus rectum $=\dfrac{2b^2}{a}=\dfrac{18}{4}=\dfrac{9}{2}$.
• Eccentricity: $b^2=a^2(1-e^2)\implies 9=16(1-e^2)\implies e^2=\dfrac{7}{16}\implies e=\dfrac{\sqrt7}{4}$.
• Centre $=(0,0)$. …