Question

Find the length of major axis, the eccentricity the latus rectum, the coordinate of the centre, the foci, the vertices and the equation of the directrices of following ellipse:
$16x^2+y^2=16.$

Answer 1

The given equation is $16x^2+y^2=16$
It can be written as
$16x^2+y^2=16$
$\frac{x^2}{1}+\frac{y^2}{16}=1$
$\frac{x^2}{1^2}+\frac{y^2}{4^2}=1...\left(1\right)$
Herethe denominator of $\frac{y^2}{4^2}$ is greater than the denominator of $\frac{x^2}{1^2}$
Therefore, the major axis is along the $y$-axis while the minor axis is along the $x$-axis.
On comparing equation (1) with $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$, we obtain $b=1$ and $a=4$
$\therefore ae=c=\sqrt{a^2-b^2}=\sqrt{16-1}=\sqrt{15}$
Therefore, the coordinates of the foci are $(0,\pm \sqrt{15})$
The coordinates of the vertices are $(0,\pm 4)$
Length of major axis $=2a=8$
Length of minor axis $=2b=2$
Eccentricity $e=$ $\frac{c}{a}=\frac{\sqrt{15}}{4}$
Length of latus rectum $=$ $\frac{2b^2}{a}=\frac{2\times 1}{4}=\frac{1}{2}$