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CBSE
Mathematics
Eccentricity of an ellipse
If the distance between foci of an ellipse is equal to the length of the latus rectum, then the eccentricity is,
A) $\dfrac{1}{4}\left( {\sqrt 5 - 1} \right)$
B) $\dfrac{1}{2}\left( {\sqrt 5 + 1} \right)$
C) $\dfrac{1}{2}\left( {\sqrt 5 - 1} \right)$
D) $\dfrac{1}{4}\left( {\sqrt 5 + 1} \right)$

CBSE
Mathematics
Eccentricity of an ellipse
If the eccentricity ${e_1}$ is of the ellipse $\dfrac{{{x^2}}}{{16}} + \dfrac{{{y^2}}}{{25}} = 1$ and ${e_2}$ is the eccentricity of the hyperbola passing through the foci of the ellipse and ${e_1} \times {e_2} = 1$, then the equation of the hyperbola. Is
A). $\dfrac{{{x^2}}}{9} - \dfrac{{{y^2}}}{{16}} = 1$
B). $\dfrac{{{x^2}}}{{16}} - \dfrac{{{y^2}}}{9} = - 1$
C). $\dfrac{{{x^2}}}{9} - \dfrac{{{y^2}}}{{25}} = 1$
D). None of these

CBSE
Mathematics
Eccentricity of an ellipse
Eccentricity of the ellipse ${{x}^{2}}+2{{y}^{2}}-2x+3y+2=0$ is:
1. $\dfrac{1}{\sqrt{2}}$
2. $\dfrac{1}{2}$
3. $\dfrac{1}{2\sqrt{2}}$
4. $\dfrac{1}{\sqrt{3}}$

CBSE
Mathematics
Eccentricity of an ellipse
An ellipse has eccentricity $\dfrac{1}{2}$ and one focus at the point $P\left( \dfrac{1}{2},1 \right)$. Its one directrix is the common tangent nearer to the point $P$ to the circle ${{x}^{2}}+{{y}^{2}}=1$ and the hyperbola ${{x}^{2}}-{{y}^{2}}=1$ . The equation of the ellipse in the standard form is
A) $\dfrac{{{\left( x-\dfrac{1}{3} \right)}^{2}}}{\dfrac{1}{9}}+\dfrac{{{\left( y-1 \right)}^{2}}}{\dfrac{1}{12}}=1$
B) $\dfrac{{{\left( x-\dfrac{1}{3} \right)}^{2}}}{\dfrac{1}{9}}+\dfrac{{{\left( y+1 \right)}^{2}}}{\dfrac{1}{12}}=1$
C) $\dfrac{{{\left( x-\dfrac{1}{3} \right)}^{2}}}{\dfrac{1}{9}}-\dfrac{{{\left( y-1 \right)}^{2}}}{\dfrac{1}{12}}=1$
D) $\dfrac{{{\left( x-\dfrac{1}{3} \right)}^{2}}}{\dfrac{1}{9}}-\dfrac{{{\left( y+1 \right)}^{2}}}{\dfrac{1}{12}}=1$
CBSE
Mathematics
Eccentricity of an ellipse
S and T are the foci of an ellipse and $B$is the endpoint of the minor axis. If $STB$ is an equilateral triangle, then the eccentricity of the ellipse is:
1. $\dfrac{1}{4}$
2. $\dfrac{1}{3}$
3. $\dfrac{1}{2}$
4. $\dfrac{2}{3}$
CBSE
Mathematics
Eccentricity of an ellipse
The equation of the ellipse whose equation of directrix is$3x + 4y - 5 = 0$, coordinates of the focus are$\left( {1,2} \right)$and the eccentricity is $\dfrac{1}{2}$is$91{x^2} + 84{y^2} - 24xy - 170x - 360y + 475 = 0$.
A. True
B. False

CBSE
Mathematics
Eccentricity of an ellipse
The length of sub tangent corresponding to the point $\left( {3,\dfrac{{12}}{5}} \right)$ on the ellipse is $\dfrac{{16}}{3}$. Then the eccentricity of the ellipse is:
(A) $\dfrac{4}{5}$
(B) $\dfrac{2}{3}$
(C) $\dfrac{1}{5}$
(D) $\dfrac{3}{5}$
CBSE
Mathematics
Eccentricity of an ellipse
The ends of major axis of an ellipse are $(5,0);(-5,0)$ and one of the foci lies on $3x-5y-9=0$, then the eccentricity of the ellipse is: -
\begin{align} & a)\,\dfrac{2}{3} \\ & b)\,\dfrac{3}{5} \\ & c)\,\dfrac{4}{5} \\ & d)\,\dfrac{1}{3} \\ \end{align}
CBSE
Mathematics
Eccentricity of an ellipse
An ellipse has eccentricity $\dfrac{1}{2}$ and one focus at the point $P\left( {\dfrac{1}{2},1} \right)$. Its one directrix is the common tangent, nearer to the point P, to the circle ${x^2} + {y^2} = 1$ and hyperbola ${x^2} - {y^2} = 1$. Find the equation of the ellipse in standard form.
CBSE
Mathematics
Eccentricity of an ellipse
The orbit of the earth is an ellipse with eccentricity $\dfrac{1}{{60}}$ with Sun at one focus, the major axis being approximately $186 \times 1{0^6}\;miles$ in length. The shortest and longest distance of the earth from the Sun is
A.$9145 \times {10^4}\;{\text{miles}}$, $9455 \times {10^4}\;{\text{miles}}$
B.$9147 \times {10^4}\;{\text{miles}}$, $9457 \times {10^4}\;{\text{miles}}$
C.$9145 \times {10^6}\;{\text{miles}}$, $9455 \times {10^6}\;{\text{miles}}$
D.None of these
CBSE
Mathematics
Eccentricity of an ellipse
An ellipse has OB as semi-minor axis, F and F’ its foci and the $\angle FBF'$ is a right angle. Then, the eccentricity of the ellipse is
(A) $\dfrac{1}{\sqrt{3}}$
(B) $\dfrac{1}{4}$
(C) $\dfrac{1}{2}$
(D) $\dfrac{1}{\sqrt{2}}$
CBSE
Mathematics
Eccentricity of an ellipse
Write the eccentricity of the ellipse $9{x^2} + 5{y^2} - 18x - 2y - 16 = 0$.

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