Tag: standard equation of an ellipse

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Questions Related to standard equation of an ellipse

Multiple choice mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

The graph of the equation $4y^2 + x^2= 25$ is

  1. a circle

  2. an ellipse

  3. a hyperbola

  4. a parabola

  5. a straight line
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given, $4{y}^{2}+{x}^{2}=25$

$\Rightarrow \dfrac { { y }^{ 2 } }{ 25/4 } +\dfrac { { x }^{ 2 } }{ 25 } =1$
It is in the form of ellipse $\left (\dfrac { { y }^{ 2 } }{ {a}^{2} } +\dfrac { { x }^{ 2 } }{ {b}^{2} } =1\right)$
So, the correct answer is option $B$.

Multiple choice mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

Latus rectum of the conic satisfying the differential equation $x dy+y dx=0$ and passing through the point $(2,8)$ is :

  1. $4\sqrt{2}$
  2. $8$
  3. $8\sqrt{2}$
  4. $16$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The differential equation is

$xdy+ydx=0$
$d(xy)=0$
By integrating we get,
$xy=k$..........(1)

Equation (1) passes through the point $(2,8)$, so
$(2)(8)=k$
$k=16$

So, equation of the conic is 
$xy=16$
which is a rectangular hyperbola $(xy=c^{2})$, where $c=4$

Length of latus rectum for rectangular hyperbola is $2\sqrt{2}c=8\sqrt{2}$
 





Multiple choice mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

The foci of an ellipse are located at the points $(2, 4)$ and $(2, -2)$. The points $(4, 2)$ lies on the ellipse. If $a$ and $b$ represent the lengths of the semi-major and semi-minor axes respectively, then the value of $(ab)^{2}$ is equal to

  1. $68 + 22\sqrt {10}$
  2. $6 + 22\sqrt {10}$
  3. $26 + 10\sqrt {10}$
  4. $6 + 10\sqrt {10}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The distance between the foci is $6$, so $c = 3$.
The sum of the distance from $(4, 2)$ to each of the foci is the major axis length,
so
$2a = \sqrt {(4 - 2)^{2} + (2 - 4)^{2}} + \sqrt {(4 - 2)^{2} + (2 + 2)^{2}}$
$= \sqrt {4 + 4} + \sqrt {4 + 16} = \sqrt {8} + \sqrt {20}$
$= 2\sqrt {2} + 2\sqrt {5} \Rightarrow a = \sqrt {2} + \sqrt {5}$
Also, for an ellipse,
$b^{2} = a^{2} - c^{2} = (\sqrt {2} + \sqrt {5})^{2} - 3^{2}$
$= 7 + 2\sqrt {10} = -2 + 2\sqrt {10}$.
Thus, we have
$(ab)^{2} = (7 + 2\sqrt {10})(-2 + 2\sqrt {10})$
$= -14 + 14\sqrt {10} - 4\sqrt {10} + 40$
$= 26 + 10\sqrt {10}$.

Multiple choice mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

Which of the following is/are not false?

  1. The mid point of the line segment joining the foci is called the centre of the ellipse.

  2. The line segment through the foci of the ellipse is called the major axis.

  3. The end points of the major axis are called the vertices of the ellipse.

  4. Ellipse is symmetric with respect to Y-axis only.
Reveal answer Fill a bubble to check yourself
A,B,C Correct answer
Explanation

(A) Midpoint of the line segment joining the foci is called the centre of ellipse: TRUE


(B) Line segment through the foci is called major axis: TRUE

(C) End point of major axis are called vertices of ellipse: TRUE

(D) Ellipse is symmetric with respect to Y-axis only.: FALSE
(Ellipse is symmetric to both x-axis and y-axis)

Multiple choice mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

The equation $2x^2+3y^2-8x-18y+35=\lambda$ represents?

  1. A circle for all $\lambda$
  2. An ellipse if $\lambda < 0$
  3. The empty set if $\lambda > 0$
  4. A-point if $\lambda = 0$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given:

$ 2x^{2} + 3y - 8x - 18y + 35 = \lambda  $

$ 2\left (x^{2} - 4x \right ) + 3 \left ( y^{2} - 6y + 35 \right ) = \lambda  $

$ 2\left (x - 2 \right )^{2} + 3 \left ( y - 3 \right )^{2} = \lambda  $

For $ \lambda = 0 $, then

$ 2\left (x - 2 \right )^{2} + 3 \left ( y - 3 \right )^{2} = 0  $

Thus, the point is $ \left ( 2,3 \right ) $.

Hence, the correct option is ‘d’.

Multiple choice mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

The equation of ellipse whose major axis is along the direction of x-axis, eccentricity is $e=2/3$

  1. $36x^2+20y^2=405$
  2. $20x^2+36y^2=405$
  3. $30x^2+22y^2=411$
  4. $22x^2+32y^2=409$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$\begin{array}{l} e=\frac { 2 }{ 3 } =\sqrt { \frac { { { a^{ 2 } }-{ b^{ 2 } } } }{ c }  }  \\ \Rightarrow { \left( { \frac { 2 }{ 3 }  } \right) ^{ 2 } }=\frac { { { a^{ 2 } }-{ b^{ 2 } } } }{ { { a^{ 2 } } } }  \\ \Rightarrow \frac { { 4{ a^{ 2 } } } }{ a } ={ a^{ 2 } }-{ b^{ 2 } } \\ \Rightarrow { b^{ 2 } }={ a^{ 2 } }-4{ a^{ 2 } }=\frac { { 5{ a^{ 2 } } } }{ 9 } \to \left( i \right)  \end{array}$
Equation of Ellipse are
$\begin{array}{l} \Rightarrow \frac { { { x^{ 2 } } } }{ { { a^{ 2 } } } } +\frac { { { y^{ 2 } } } }{ { { b^{ 2 } } } } =1 \\ \Rightarrow \frac { { { x^{ 2 } } } }{ { { a^{ 2 } } } } +\frac { { 9{ y^{ 2 } } } }{ { 5{ a^{ 2 } } } } =1\to \left( { ii } \right)  \\ Put\, \, { a^{ 2 } }=\frac { { 405 } }{ { 20 } } \, \, \left( { From\, \, option\, \, in\, \, equation\left( i \right)  } \right)  \\ Then,\, \, { b^{ 2 } }=\frac { { 405 } }{ { 360 } }  \end{array}$
Hence, equation of ellipse is
$ \Rightarrow \frac{{{x^2}}}{{\left( {\frac{{405}}{{20}}} \right)}} + \frac{{{y^2}}}{{\left( {\frac{{405}}{{36}}} \right)}} =  - 1,20{x^2} + 36{y^2} = 405$
Multiple choice mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

Eccentricity of ellipse $\frac{{{x^2}}}{{{a^2} + 1}} + \frac{{{y^2}}}{{{a^2} + 2}} = 1\,is\,\frac{1}{{\sqrt 3 }}$ then length of Latus rectum is 

  1. $\frac{2}{{\sqrt 3 }}$
  2. $\frac{4}{{\sqrt 3 }}$
  3. $2\sqrt 3 $
  4. $\frac{{\sqrt 3 }}{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Let ${ A }^{ 2 }={ a }^{ 2 }+1$,  ${ B }^{ 2 }={ a }^{ 2 }+2$    {Here, ${ B }^{ 2 }>{ A }^{ 2 }$}
So, $e=\sqrt { 1-\dfrac { { A }^{ 2 } }{ { B }^{ 2 } }  } =\sqrt { 1-\dfrac { { a }^{ 2 }+1 }{ { a }^{ 2 }+2 }  } =\dfrac { 1 }{ \sqrt { { a }^{ 2 }+2 }  } =\dfrac { 1 }{ \sqrt { 3 }  } $
So,  $\sqrt { { a }^{ 2 }+2 } =\sqrt { 3 } \Rightarrow a=\pm 1$
So, length of lotus return $=2\dfrac { { A }^{ 2 } }{ B } $
Length $=\dfrac { 2\left( { a }^{ 2 }+1 \right)  }{ \sqrt { { a }^{ 2 }+2 }  } =\dfrac { 2\left( 2 \right)  }{ \sqrt { 3 }  } =\dfrac { 4 }{ \sqrt { 3 }  } $
Multiple choice mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

If the latus rectum of an ellipse $x ^ { 2 } \tan ^ { 2 } \varphi + y ^ { 2 } \sec ^ { 2 } \varphi =$ $1$ is $1 / 2 ,$ then $\varphi$ is

  1. $\pi / 2$
  2. $\pi / 6$
  3. $\pi / 3$
  4. $5$ $\pi/ 12$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given $x^2 tan^2 \phi + y^2 \, sec^2 \phi = 1$

$\rightarrow \dfrac{x^2}{(1/tan^2 \phi)} + \dfrac{y^2}{(1/sec^2 \phi)} = 1$
$a = \pm \dfrac{1}{tan \phi} , b = \pm \dfrac{1}{sec \phi}$
and $\rightarrow e^2 = 1 - \dfrac{b^2}{a^2}$
$\rightarrow e^2 = 1 - \dfrac{1/sec^2 \phi}{1/tan^2 \phi} = 1 - \dfrac{tan^2 \phi}{sec^2 \phi}$
$\rightarrow e^2 = 1 - sin^2 \phi = cos^2 \phi$
length of latus rectum 
$(LL') = \dfrac{2 b^2}{a} = 2a (1 - e^2)$
$\rightarrow 2a (1 - cos^2 \phi) = 2a. sin^2 \phi = \dfrac{1}{2} $ (Given)
$\therefore 2. \dfrac{cos \phi}{sin \phi} sin^2 \phi = \dfrac{1}{2}$
$\rightarrow 2 cos \phi \, sin \phi = \dfrac{1}{2}$
$\rightarrow sin^2 \phi = \dfrac{1}{2} $
$\rightarrow 2 \phi = \dfrac{\pi}{6} , \dfrac{5 \pi}{6}$
$\therefore \phi = \dfrac{\pi}{12}$    or 
$\phi = \dfrac{ 5 \pi}{12}$

Multiple choice mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

The curve represented by $Rs \left(\dfrac{1}{z}\right)=C$ is (where $C$ is a constant and $\neq 0$)

  1. Ellipse

  2. Parabola

  3. Circle

  4. Straight line
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\begin{array}{l} { { Re } }\, \, \left( { \frac { 1 }{ z }  } \right) =c \ { { Re } }\, \, \left( { \frac { 1 }{ { x+iy } }  } \right) =c \ { { Re } }\, \, \left( { \frac { { x-iy } }{ { { x^{ 2 } }+{ y^{ 2 } } } }  } \right) =c \ \frac { x }{ { { x^{ 2 } }+{ y^{ 2 } } } } =c \ c\left( { { x^{ 2 } }+{ y^{ 2 } } } \right) -{ x }=0 . \end{array}$


Hence, this is represent circle.