Tag: standard equation of ellipse

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Questions Related to standard equation of ellipse

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

The equation of the tangent to the ellipse such that sum of perpendiculars dropped from foci is 2 units, is

  1. $y cos3\pi/ 4 - x sin 3\pi /4=1$
  2. $y sin \frac{3\pi}{8}- x cos \frac{3\pi}{8}=1$
  3. $x cos \pi /8 - sin \pi /8=1$
  4. $y cos \frac{5\pi}{8}+x sin \frac{5\pi}{8}=1$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The product of perpendiculars from foci to any tangent of an ellipse is b^2. The sum of perpendiculars is a property related to the tangent equation. The given option A fits the standard form.

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

An ellipse $\cfrac { { x }^{ z } }{ 4 } +\cfrac { { y }^{ z } }{ 3 } =1$ confocal with hyperbola $\cfrac { { x }^{ 2 } }{ \cos ^{ 2 }{ \theta  }  } -\cfrac { { y }^{ 2 } }{ \sin ^{ 2 }{ \theta  }  } =1$ then the set of value of $'0'$

  1. $R$
  2. $R-\left\{ n\pi ,n\epsilon z \right\} $
  3. $R-\left\{ \left( 2n+1 \right) \cfrac { \pi }{ 2 } ,n\epsilon z \right\} $
  4. $R-\left\{ \cfrac { n\pi }{ 2 } ,n\epsilon z \right\} $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Focus of ellipse$=ae=a\sqrt { 1-\cfrac { { b }^{ 2 } }{ { a }^{ 2 } }  } $
$=\sqrt { { a }^{ 2 }-{ b }^{ 2 } } $
$=\sqrt { 1 } $
$=1$
Focus of hyperbola$=a\sqrt { 1+\cfrac { { b }^{ 2 } }{ { a }^{ 2 } }  } $
$=\sqrt { { a }^{ 2 }+{ b }^{ 2 } } $
$=\sqrt { \sin ^{ 2 }{ \theta  } +\cos ^{ 2 }{ \theta  }  } $
$=\sqrt { 1 } $
$=1$
$\therefore $The ellipse and hyperbola will be confocal for $\theta \epsilon R$.
Multiple choice mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

Equation of the ellipse whose axes are the axes of coordinates and which passes through the point $ (-3,1)$ and has eccentricity $\sqrt {\frac{2}{5}} $ is 

  1. $5x^3+3y^2-48=0$
  2. $3x^2+5y^2-15=0$
  3. $5x^2+3y^2-32=0$
  4. $3x^2+5y^2-32=0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

We know that equation of ellipse is

 

  $ \dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}=1 $       …….(1)

Given that

  $ e=\sqrt{\dfrac{2}{5}} $

 $ \sqrt{\dfrac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}}}=\sqrt{\dfrac{2}{5}} $

 

Taking square both side and solving , we get


  $ 5{{a}^{2}}-5{{b}^{2}}=2{{a}^{2}} $

 $ {{a}^{2}}=\dfrac{5{{b}^{2}}}{3} $    …….(2)

$\because $ ellipse pass through (-3,1)

Then $x=-3, y=1$

Put in equation (1) we get

  $ \dfrac{{{\left( -3 \right)}^{2}}}{{{a}^{2}}}+\dfrac{{{1}^{2}}}{{{b}^{2}}}=1 $

 $ {{a}^{2}}+9{{b}^{2}}={{a}^{2}}{{b}^{2}} $

 $ \dfrac{5{{b}^{2}}}{3}+9{{b}^{2}}=\dfrac{5{{b}^{2}}}{3}.{{b}^{2}} $

 $ {{b}^{2}}=\dfrac{32}{5} $    (From equation (1) and (2)  )

Put in equation (2) , we get ${{a}^{2}}=\dfrac{32}{3}$

the value of a and b put in equation (1), we get


  $ \dfrac{{{x}^{2}}}{\dfrac{32}{3}}+\dfrac{{{y}^{2}}}{\dfrac{32}{5}}=1 $

 $ 3{{x}^{2}}+5{{y}^{2}}=32 $

This is required equation

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

S and S' foci of an ellipse. B is one end of the minor axis. If $\angle{SBS'}$ is a right angled isosceles triangle, then e$=?$

  1. $\dfrac{1}{\sqrt{2}}$
  2. $\dfrac{1}{2}$
  3. $\dfrac{\sqrt{3}}{2}$
  4. $\dfrac{3}{4}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
We have
$S=(ae,0)\quad S'(-ae,0)and B=(0,b)$
Since it is given that $\angle SBS'=90^o$
Slope of SB$\times$ Slope of S'B$=-1$
$\left(\dfrac{b-0}{0-ae}\right)\times\left(\dfrac{b-0}{b+ae}\right)=-1$
$\left(\dfrac{-b}{ae}\right)\left(\dfrac{b}{ae}\right)=-1$
$b^2=a^2e^2$
But, $b^2=a^2(1-e^2)$
So,
$a^2(1-e^2)=a^2e^2$
$1-e^2=e^2$
$2e^2=1$
$e^2=\dfrac{1}{2}$
$e=\dfrac{1}{\sqrt2}$
Multiple choice mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

The eccentricity of an ellipse is $\dfrac {\sqrt {3}}{2}$ its length of latus reetum is

  1. $\dfrac {1}{2}$ (length of major axis)
  2. $\dfrac {1}{3}$ (length of major axis)
  3. $\dfrac {1}{4}$ (length of major axis)
  4. $\dfrac {2}{3}$ (length of major axis)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Latus rectum L = 2b^2/a. Given e = sqrt(3)/2, e^2 = 3/4. Since b^2 = a^2(1-e^2), b^2 = a^2(1/4). So L = 2(a^2/4)/a = a/2. The length of the major axis is 2a. Thus, L = (a/2) / (2a) = 1/4 of the major axis.

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

The length of latus rectum of $\dfrac {x^2}9+\dfrac {y^2}2=1$ is 

  1. $\dfrac 74$
  2. $\dfrac 34$
  3. $\dfrac 43$

  4. None.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The length of latus Rectum of $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$

is $\dfrac{2b^2}{a}$

Here $a=3\quad b=\sqrt 2$

$\Rightarrow \dfrac{2b^2}{a}=\dfrac{2(\sqrt 2)^2}{3}=\dfrac{2(2)}{3}=\dfrac 43$

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

An ellipse of semi-axis $a,b,$ slides between two perpendicular lines, then the locus of its foci is, (the two lines being taken  as the axes of coordinates)

  1. $(x^{2}+y^{2})(x^{2}y^{2}+b^{2})=4a^{2}x^{2}y^{2}$
  2. $(x^{2}+y^{2})(x^{2}y^{2}+b^{2})=4b^{2}x^{2}y^{2}$
  3. $(x^{2}-y^{2})(x^{2}y^{2}+b^{2})=4b^{2}x^{2}y^{2}$
  4. $(x^{2}-y^{2})(x^{2}y^{2}+b^{2})=4a^{2}x^{2}y^{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

This is a classic locus problem involving an ellipse sliding between two perpendicular axes. The locus of the foci is derived using coordinate geometry.

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

If equation $(5x-1)^{2}+(5y-2)^{2}=(\lambda^{2}-2\lambda+1)(3x+4y-1)^{2}$ represents an ellipse, then $\lambda \in$

  1. $(0, 1)$
  2. $(0, 2)$
  3. $(1, 2)$
  4. $(0, 1)\cup (1, 2)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The equation represents an ellipse if the eccentricity e < 1. This condition relates to the coefficients of the quadratic form.

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

The equation $\dfrac{{x}^{2}}{2-r}+\dfrac{{y}^{2}}{r-5}+1=0$ represents an ellipse if

  1. $r>1$
  2. $r>5$
  3. $2 < r< 5$
  4. $r<2$ or $r>5$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For the equation to represent an ellipse, the denominators must be positive and the coefficients must allow for the standard form x^2/A + y^2/B = 1. This requires 2-r > 0 and r-5 < 0, or vice versa, leading to 2 < r < 5.