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 $\frac{x^{2}}{25}+\frac{y^{2}}{100}=1$
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 $\frac{x^{2}}{25}+\frac{y^{2}}{100}=1$
The given equation is $\frac{x^{2}}{25}+\frac{y^{2}}{100}=1$ or $\frac{x^{2}}{5^{2}}+\frac{y^{2}}{10^{2}}=1$
Here, the denominator of $\frac{y^{2}}{100}$ is greater than the denominator of $\frac{x^{2}}{25}$
Therefore, the major axis is along the $y-$ axis, while the minor axis is along the $x-$ axis.
On comparing the given equation with $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1,$ we obtain $b=5$ and $a=10$
$\therefore c=\sqrt{a^{2}-b^{2}}=\sqrt{100-25}=\sqrt{75}=5 \sqrt{3}$
Therefore,
The coordinates of the foci are $(0, \,\pm 5 \sqrt{3})$
The coordinates of the vertices are $(0,\,±10)$
Length of major axis $=2 a=20$
Length of minor axis $=2 b=10$
Eccentricity, $e=\frac{c}{a}=\frac{5 \sqrt{3}}{10}=\frac{\sqrt{3}}{2}$
Length of latus rectum $=\frac{2 b^{2}}{a}=\frac{2 \times 25}{10}=5$
Similar Questions
The sum of the focal distances of any point on the conic $\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{{16}} = 1$ is
The sum of the focal distances of any point on the conic $\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{{16}} = 1$ is
The number of real tangents that can be drawn to the ellipse $3x^2 + 5y^2 = 32$ passing through $(3, 5)$ is
The number of real tangents that can be drawn to the ellipse $3x^2 + 5y^2 = 32$ passing through $(3, 5)$ is
An ellipse $\frac{\left(x-x_0\right)^2}{a^2}+\frac{\left(y-y_0\right)^2}{b^2}=1$, $a > b$, is tangent to both $x$ and $y$ axes and is placed in the first quadrant. Let $F_1$ and $F_2$ be two foci of the ellipse and $O$ be the origin with $OF _1 < OF _2$. Suppose the triangle $OF _1 F _2$ is an isosceles triangle with $\angle OF _1 F _2=120^{\circ}$. Then the eccentricity of the ellipse is
An ellipse $\frac{\left(x-x_0\right)^2}{a^2}+\frac{\left(y-y_0\right)^2}{b^2}=1$, $a > b$, is tangent to both $x$ and $y$ axes and is placed in the first quadrant. Let $F_1$ and $F_2$ be two foci of the ellipse and $O$ be the origin with $OF _1 < OF _2$. Suppose the triangle $OF _1 F _2$ is an isosceles triangle with $\angle OF _1 F _2=120^{\circ}$. Then the eccentricity of the ellipse is
- [KVPY 2021]
An ellipse is drawn with major and minor axes of lengths $10 $ and $8$ respectively. Using one focus as centre, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. The radius of the circle is
An ellipse is drawn with major and minor axes of lengths $10 $ and $8$ respectively. Using one focus as centre, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. The radius of the circle is
Let $\mathrm{A}(\alpha, 0)$ and $\mathrm{B}(0, \beta)$ be the points on the line $5 x+7 y=50$. Let the point $P$ divide the line segment $A B$ internally in the ratio $7: 3$. Let $3 x-$ $25=0$ be a directrix of the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and the corresponding focus be $S$. If from $S$, the perpendicular on the $\mathrm{x}$-axis passes through $\mathrm{P}$, then the length of the latus rectum of $\mathrm{E}$ is equal to
Let $\mathrm{A}(\alpha, 0)$ and $\mathrm{B}(0, \beta)$ be the points on the line $5 x+7 y=50$. Let the point $P$ divide the line segment $A B$ internally in the ratio $7: 3$. Let $3 x-$ $25=0$ be a directrix of the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and the corresponding focus be $S$. If from $S$, the perpendicular on the $\mathrm{x}$-axis passes through $\mathrm{P}$, then the length of the latus rectum of $\mathrm{E}$ is equal to
- [JEE MAIN 2024]