Tag: standard equation of ellipse

Questions Related to standard equation of ellipse

Find the the length of the major axis of ellipse: $12x^2+4y^2+24x-16y+25=0$

mathematics and statistics conic sections standard equation of ellipse introduction to ellipse standard equation of an ellipse

  1. $ \sqrt{3}$

  2. $ 2 \sqrt{3}$

  3. $ 2 \sqrt{\dfrac32}$

  4. $ 2 \sqrt{6}$

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Correct Option: A
Explanation:

$12x^{ 2 }+4y^{ 2 }+24x-16y+25=0$


$\Rightarrow 12(x^2+2x+1)+4(y^2-4y+4)=3$

$\Rightarrow 12(x+1)^2+4(y-2)^2=(\sqrt 3)^2$

$\Rightarrow \displaystyle \dfrac { { \left( x+1 \right)  }^{ 2 } }{ { \left( \dfrac { 1 }{ 2 }  \right)  }^{ 2 } } +\dfrac { { \left( y-2 \right)  }^{ 2 } }{ { { \left( \dfrac { \sqrt3 }{ 2 }  \right)  }^{ 2 } } } =1\ $


Length of major axis $=2a=\sqrt3.$

Arrange the following ellipses in the ascending order of their lengths of major axis:
$\mathrm{A}:x^{2}+2y^{2}-4x+12y+14=0$
$\displaystyle \mathrm{B}:\frac{(x-1)^{2}}{9}+\frac{(y-1)^{2}}{16}=1$
$\mathrm{C}:4x^{2}+9y^{2}=1$
$\mathrm{D}:x=3+6\cos\theta,y=5+7\sin\theta$

mathematics and statistics conic sections standard equation of ellipse introduction to ellipse standard equation of an ellipse

  1. C, A, B, D

  2. C, A, D, B

  3. A, B, C, D

  4. C, D, A, B

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Correct Option: A
Explanation:
For A:
${ x }^{ 2 }+2{ y }^{ 2 }−4x+12y+14=0\\ \Rightarrow{ x }^{ 2 }-4x+4+2\left( { y }^{ 2 }+6y+9 \right) -8=0\\ \Rightarrow { \left( x-2 \right)  }^{ 2 }+2{ \left( y+3 \right)  }^{ 2 }=8\\ \\ \Rightarrow \cfrac { { \left( x-2 \right)  }^{ 2 } }{ 8 } +\cfrac { { \left( y+3 \right)  }^{ 2 } }{ 4 } =1$

$\therefore$ Length of major axis= $2\sqrt { 2 } $

For B:
$\\ \\  \cfrac { { \left( x-1 \right)  }^{ 2 } }{ 9 } +\cfrac { { \left( y-1 \right)  }^{ 2 } }{ 16 } =1$

$\therefore$ Length of major axis= 4

For C:
$4{ x }^{ 2 }+9{ y }^{ 2 }=1\\ \Rightarrow\cfrac { { x }^{ 2 } }{ \frac { 1 }{ 4 }  } +\cfrac { { y }^{ 2 } }{ \frac { 1 }{ 9 }  } =1$

$\therefore$ Length of major axis= $\cfrac{1}{2}$

For D:
$\\ x-3=6\cos \theta\\ y-5=7\sin \theta$

$\therefore$ Length of major axis= 7

$\therefore$ Option [A] C,A,B,D

lf $ax^{2}+by^{2}+2gx+2fy+c=0$ represents an ellipse, then

mathematics and statistics conic sections standard equation of ellipse introduction to ellipse standard equation of an ellipse

  1. its major axis is parallel to $x-$axis

  2. its major axis is parallel to $y-$axis

  3. its axes (i.e. major axis and minor axis) are neither parallel to $x-$axis nor parallel to $y-$axis

  4. its axes are parallel to co-ordinate axes

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Correct Option: D

The abscissa of the focii of the ellipse $25(\mathrm{x}^{2}-6\mathrm{x}+9)+16\mathrm{y}^{2}=400$ is:

mathematics and statistics conic sections standard equation of ellipse introduction to ellipse standard equation of an ellipse

  1. $ (4,-ae), (4,ae)$

  2. $ (3,-ae), (3,ae)$

  3. $ (5,-ae), (5,ae)$

  4. None of these

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Correct Option: B
Explanation:

The equation of the ellipse can be written as $\dfrac{(x-3)^{2}}{4^{2}}+\dfrac{y^{2}}{5^{2}}=1$
$\therefore $ Minor axis is along the line $x-3=0$ and major axis is $y=0$ 
$(\because a^{2}=4^{2}< b^{2}=5^{2})$
$\therefore S\,$ and $ S^{'}$ are $(3,3),(3,-3)$
$\because 4^{2}=5^{2}(1-e^{2})\Rightarrow e=\dfrac{3}{5}\left ( \because ae=5\dfrac{3}{5}=3 \right )$ and $S\equiv(3,ae )$, $S^{'}\equiv(3,-ae).$
Ans: B

The eccentricity of the curve with equation ${ x }^{ 2 }+{ y }^{ 2 }-2x+3y+2=0$ is

mathematics and statistics conic sections standard equation of ellipse introduction to ellipse standard equation of an ellipse

  1. $0$

  2. $\sqrt { 2 }$

  3. $1/2$

  4. ${ 1 }/{ \sqrt { 2 } }$

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Correct Option: A
Explanation:

Given ${x}^{2}+{y}^{2}-2x+3y+2=0$

$\Rightarrow \left({x}^{2}-2x\right)+\left({y}^{2}+3y\right)+2=0$
$\Rightarrow \left({x}^{2}-2x+1-1\right)+\left({y}^{2}+2\times 1\times \dfrac{3}{2}+\dfrac{9}{4}-\dfrac{9}{4}\right)+2=0$
$\Rightarrow {\left(x-1\right)}^{2}-1+{\left(y+\dfrac{3}{2}\right)}^{2}-\dfrac{9}{4}+2=0$
$\Rightarrow {\left(x-1\right)}^{2}+{\left(y+\dfrac{3}{2}\right)}^{2}=-2+1+\dfrac{9}{4}=\dfrac{5}{4}$
Divide both sides by ${\left(\sqrt{\dfrac{5}{4}}\right)}^{2}$ we get
$\dfrac{{\left(x-1\right)}^{2}}{{\left(\sqrt{\dfrac{5}{4}}\right)}^{2}}+\dfrac{{\left(y+\dfrac{3}{2}\right)}^{2}}{{\left(\sqrt{\dfrac{5}{4}}\right)}^{2}}$  is an ellipse where $a=\sqrt{\dfrac{5}{4}}$ and  $b=\sqrt{\dfrac{5}{4}}$
Eccentricity $e=\sqrt{1-\dfrac{{b}^{2}}{{a}^{2}}}=\sqrt{1-\dfrac{{\left(\sqrt{\dfrac{5}{4}}\right)}^{2}}{{\left(\sqrt{\dfrac{5}{4}}\right)}^{2}}}=\sqrt{1-1}=0$