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# 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}{49}$$+\large\frac{y^2}{36}$$=1$

$\begin {array} {1 1} (A)\;Axis : Major \: axis \: along \: x - axis , Foci : ( \pm 13, 0), Vertices : ( \pm 7, 0) , Length \: of \: the \: latus \: rectum : 72 / 7, Length \: of \: the \: minor \: axis : 12, length \: of \: the \: major \: axis : 14, eccentricity \: e : 13/7 \\ (B)\;Axis : Major \: axis \: along \: y - axis , Foci : (0, \pm 13), Vertices : ( 0, \pm 7) , Length \: of \: the \: latus \: rectum : 72 / 7, Length \: of \: the \: minor \: axis : 12, length \: of \: the \: major \: axis : 14, eccentricity \: e : 13/7 \\ (C)\;Axis : Major \: axis \: along \: x - axis , Foci : ( \pm 13, 0), Vertices : ( \pm 7, 0) , Length \: of \: the \: latus \: rectum : 72 / 7, Length \: of \: the \: minor \: axis : 12, length \: of \: the \: major \: axis : 14, eccentricity \: e : \sqrt{ 13}/7 \\ (D)\;Axis : Major \: axis \: along \: y - axis , Foci : ( 0, \pm 13), Vertices : ( 0,\pm 7) , Length \: of \: the \: latus \: rectum : 72 / 7, Length \: of \: the \: minor \: axis : 12, length \: of \: the \: major \: axis : 14, eccentricity \: e : \sqrt{13}/7 \end {array}$

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}{49}$$+\large\frac{y^2}{36}$$=1$
Comparing this equation with the equation of ellipse $\large\frac{x^2}{a^2}$$+\large\frac{y^2}{b^2}$$=1$ we get,
$a^2=49$ and $b^2=36$
$\therefore$ The major axis is along x - axis and the minor axis is along y - axis.
Hence $c = \sqrt{a^2-b^2}$
$= \sqrt{49-36}$
$= \sqrt{13}$
Step 2 :
$\therefore$ The coordinates of foci are $( \pm \sqrt{13}, 0)$
The coordinates of vertices are $(\pm 7, 0)$
Length of major axis = 2a = 14
Length of the minor axis = 2b = 12