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 $\frac{x^{2}}{16}+\frac {y^2} {9}=1$.
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 $\frac{x^{2}}{16}+\frac {y^2} {9}=1$.
The given equation is $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ or $\frac{x^{2}}{4^{2}}+\frac{y^{2}}{3^{2}}=1$
Here, the denominator of $\frac{x^{2}}{16}$ is greater than the denominator of $\frac{y^{2}}{9}$.
Therefore, the major axis is along the $x-$ axis, while the minor axis is along the $y-$ axis.
On comparing the given equation with $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,$ we obtain $a=4$ and $b=3$
$\therefore c=\sqrt{a^{2}-b^{2}}=\sqrt{16-9}=\sqrt{7}$
Therefore,
The coordinates of the foci are $(\pm \sqrt{7}, \,0)$
The coordinates of the vertices are $(±4,\,0)$
Length of major axis $=2 a=8$
Length of minor axis $=2 b=6$
Eccentricity, $e=\frac{c}{a}=\frac{\sqrt{7}}{4}$
Length of latus rectum $=\frac{2 b^{2}}{a}=\frac{2 \times 9}{4}=\frac{9}{2}$
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- [JEE MAIN 2021]
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Statement $-1$ : If two tangents are drawn to an ellipse from a single point and if they are perpendicular to each other, then locus of that point is always a circle
Statement $-2$ : For an ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ , locus of that point from which two perpendicular tangents are drawn, is $x^2 + y^2 = (a + b)^2$ .
Statement $-1$ : If two tangents are drawn to an ellipse from a single point and if they are perpendicular to each other, then locus of that point is always a circle
Statement $-2$ : For an ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ , locus of that point from which two perpendicular tangents are drawn, is $x^2 + y^2 = (a + b)^2$ .