Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse : $\large\frac{x^2}{16}$$+\large\frac{y^2}{9}$$=1$

$\begin {array} {1 1} (A)\;Axis : major \: axis \: is \: along \: x - axis, Foci : ( \pm 7, 0), Vertices : ( \pm 4, 0) , Length \: of \: the \: major \: axis = 8, Length \: of \: the minor \: axis = 6, eccentricity = \large\frac{\sqrt 7}{4} \\ (B)\;Axis : major \: axis \: is \: along \: y - axis, Foci : (0, \pm 7), Vertices : ( 0, \pm 4) , Length \: of \: the \: major \: axis = 8, Length \: of \: the minor \: axis = 6, eccentricity = \large\frac{\sqrt 7}{4} \\ (C)\;Axis : major \: axis \: is \: along \: x - axis, Foci : ( \pm2 \sqrt 7, 0), Vertices : ( \pm 4, 0) , Length \: of \: the \: major \: axis = 8, Length \: of \: the minor \: axis = 6, eccentricity = \large\frac{\sqrt 7}{2} \\ (D)\;Axis : major \: axis \: is \: along \: y - axis, Foci : (0, \pm 2\sqrt 7), Vertices : (0, \pm 4) , Length \: of \: the \: major \: axis = 8, Length \: of \: the minor \: axis = 6, eccentricity = \large\frac{\sqrt 7}{2} \end {array}$

Toolbox:
• Equation of an ellipse along major axis is $\large\frac{x^2}{a^2}$$+\large\frac{y^2}{b^2}$$=1$
• $c = \sqrt{a^2-b^2}$, where c is the focus of the ellipse.
• Coordinates of vertices are $( \pm a, 0)$
• Length of the latus rectum is $\large\frac{2b^2}{a}$
• Eccentricity $e=\large\frac{c}{a}$
• Length of the major axis is 2a ; Length of the minor axis is 2b.
Step 1 :
The given equation is $\large\frac{x^2}{16}$$+\large\frac{y^2}{9}$$=1$
Here $a^2 > b^2$
$\Rightarrow a > b$
$\therefore$ The major axis is along x - axis and the minor axis is along y - axis.
On comparing this with the equation of elipse
$\large\frac{x^2}{a^2}$$+\large\frac{y^2}{b^2}$$=1$
$a=4$ and $b=3$
$\therefore c = \sqrt{a^2-b^2} = \sqrt{16-9}$
Hence the coordinates of the foci are
$( \pm 7, 0)$
Step 2 :
Coordinates of vertices are $(\pm 4, 0)$
Length of the major axis is $2a=2 \times 4 = 8$
Length of the minor axis is $2b=2 \times 3 = 6$
Step 3 :
Eccentricity $e = \large\frac{c}{a}$$=\large\frac{\sqrt 7}{4} Length of the latus rectum is \large\frac{2b^2}{a} = \large\frac{2 \times 9}{4}$$= \large\frac{9}{2}$
edited Jul 15, 2014