Question

Find the coordinatesof the foce, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
$16x^2+y^2=16$

Answer 1

the equation fo given ellipse is
$16x^2+y^2=16$
i.e. $\frac{16x^2}{16}+\frac{y^2}{16}=1\Rightarrow \frac{x^2}{1}+\frac{y^2}{16}=1\text{Now}16>1\Rightarrow a^2=16and{b}^2=1\text{So teh equation of ellipse in standard form is}\frac{y^2}{a^2}+\frac{x^2}{b^2}=1\therefore a^2=16\Rightarrow a=4and{b}^2=1\Rightarrow 1\text{We know that}c=\sqrt{a^2-b^2}\therefore c=\sqrt{16-1}=\sqrt{15}\therefore \text{Coordinates of foci}(0,\pm c)i.e.(0,\pm \sqrt{15})\text{Coordinates fo vertices are}(0,\pm a)i.e.(0,\pm 4)\text{Length of major axis}=2a=2\times 4=8\text{Legnth of minor axis}=2b=2\times 1=2\text{Eccentricity}\left(e\right)=\frac{c}{a}=\frac{\sqrt{15}}{4}\text{Length of latus rectum}=\frac{2b^2}{a}=\frac{2\times 1}{4}=\frac{1}{2}$