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GradeEccentricity of an Ellipse

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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) $\dfrac{1}{4}\left( {\sqrt 5 + 1} \right)$

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) $\dfrac{1}{4}\left( {\sqrt 5 + 1} \right)$

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

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

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. $\dfrac{1}{\sqrt{3}}$

1. $\dfrac{1}{\sqrt{2}}$

2. $\dfrac{1}{2}$

3. $\dfrac{1}{2\sqrt{2}}$

4. $\dfrac{1}{\sqrt{3}}$

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) \[\dfrac{{{\left( x-\dfrac{1}{3} \right)}^{2}}}{\dfrac{1}{9}}-\dfrac{{{\left( y+1 \right)}^{2}}}{\dfrac{1}{12}}=1\]

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) \[\dfrac{{{\left( x-\dfrac{1}{3} \right)}^{2}}}{\dfrac{1}{9}}-\dfrac{{{\left( y+1 \right)}^{2}}}{\dfrac{1}{12}}=1\]

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. \[\dfrac{2}{3}\]

1. \[\dfrac{1}{4}\]

2. \[\dfrac{1}{3}\]

3. \[\dfrac{1}{2}\]

4. \[\dfrac{2}{3}\]

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

A. True

B. False

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) $\dfrac{3}{5}$

(A) $\dfrac{4}{5}$

(B) $\dfrac{2}{3}$

(C) $\dfrac{1}{5}$

(D) $\dfrac{3}{5}$

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}$

$\begin{align}

& a)\,\dfrac{2}{3} \\

& b)\,\dfrac{3}{5} \\

& c)\,\dfrac{4}{5} \\

& d)\,\dfrac{1}{3} \\

\end{align}$

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.

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

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

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) \[\dfrac{1}{\sqrt{2}}\]

(A) \[\dfrac{1}{\sqrt{3}}\]

(B) \[\dfrac{1}{4}\]

(C) \[\dfrac{1}{2}\]

(D) \[\dfrac{1}{\sqrt{2}}\]

Write the eccentricity of the ellipse $9{x^2} + 5{y^2} - 18x - 2y - 16 = 0$.

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