Tag: ellipse

Quizzes (5) Questions (224)

Questions Related to ellipse

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

The total number of real tangents that can be drawn to the ellipse $3x^{2}+5y^{2}=32$ and $25x^{2}+9y^{2}=450$ passing through $(3,5)$ is

  1. $0$
  2. $2$
  3. $3$
  4. $4$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$(3,5)$ lies on $25x^2+9y^2=450$

Therefore, one tangent can be drawn

and $(3,5)$ lies outside $3x^2+5y^2=32$ because $S _1>0$

Therefore, two tangents can be drawn.
So total 3 tangents

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

$\mathrm{S}$ and $\mathrm{S}^{'}$ are the foci of the ellipse $25x^{2}+16y^{2}=1600$, then the sum of the distances from $\mathrm{S}$ and $\mathrm{S}'$ to the point $(4\sqrt{3},5)$ is:

  1. $20$
  2. $15$
  3. $40$
  4. $30$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given ellipse is $25{ x }^{ 2 }+16{ y }^{ 2 }=1600$

This can be written as $\displaystyle \frac { { x }^{ 2 } }{ 64 } +\frac { { y }^{ 2 } }{ 100 } =1$
Compare it to standard form of ellipse to get $a=8,b=10$
here $a<b$
So eccentricity e$=\sqrt { 1-\displaystyle \frac { { a }^{ 2 } }{ { b }^{ 2 } }  } =\sqrt { 1-\displaystyle\frac { 64 }{ 100 }  } =\displaystyle \frac { 3 }{ 5 } $
Now the foci are given by $\left( 0,\pm \sqrt { { b }^{ 2 }-{ a }^{ 2 } }  \right) $
So $foci:\left( 0,\pm \sqrt { 36 }  \right) $
$S:\left( 0,6 \right) ,{ S } _{ 1 }:\left( 0,-6 \right) $
Now find distance of $S$ and ${ S } _{ 1 }$ from the given point $\left( 4\sqrt { 3 } ,5 \right) $
So sum$=\sqrt { { \left( 4\sqrt { 3 }  \right)  }^{ 2 }+1 } +\sqrt { { \left( 4\sqrt { 3 }  \right)  }^{ 2 }+{ \left( 11 \right)  }^{ 2 } } =7+13=20$

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

The length of the latusrectum of the parabola $169\left{ { \left( x-1 \right)  }^{ 2 }+{ \left( y-3 \right)  }^{ 2 } \right} ={ \left( 5x-12y+17 \right)  }^{ 2 }$

  1. $\cfrac { 14 }{ 13 } $
  2. $\cfrac { 28 }{ 13 } $
  3. $\cfrac { 12 }{ 13 } $

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Here ${ \left( x-1 \right)  }^{ 2 }+{ \left( y-3 \right)  }^{ 2 }={ \left{ \cfrac { 5x-12y+17 }{ \sqrt { { 5 }^{ 2 }+{ \left( -12 \right)  }^{ 2 } }  }  \right}  }^{ 2 }$


$\therefore$ The focus is $(1,3)$ and the directrix is $5x-12y+17=0$

The distance of the focus from the directrix

$=\left| \cfrac { 5\times 1-12\times 3+17 }{ \sqrt { { 5 }^{ 2 }+{ \left( -12 \right)  }^{ 2 } }  }  \right| =\cfrac { 14 }{ 13 } $

$\therefore$ Length of latusrectum $=2\times \cfrac { 14 }{ 13 } =\cfrac { 28 }{ 13 } $

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

The equation of the ellipse having vertices at $\displaystyle \left( \pm 5,0 \right) $ and foci $\displaystyle \left( \pm 4,0 \right) $ is

  1. $\displaystyle \frac { { x }^{ 2 } }{ 25 } +\frac { { y }^{ 2 } }{ 16 } =1$
  2. $\displaystyle 9{ x }^{ 2 }+25{ y }^{ 2 }=225$
  3. $\displaystyle \frac { { x }^{ 2 } }{ 9 } +\frac { { y }^{ 2 } }{ 25 } =1$
  4. $\displaystyle 4{ x }^{ 2 }+5{ y }^{ 2 }=20$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The vertices and foci of an ellipse are $\displaystyle \left( \pm 5,0 \right) $ and $\displaystyle \left( \pm 4,0 \right) $ respectively.
$\displaystyle \therefore \quad a=5$ and $\displaystyle ae=4$
$\displaystyle \Rightarrow \quad e=\frac { 4 }{ 5 } $
We know that,
$\displaystyle e=\sqrt { 1-\frac { { b }^{ 2 } }{ { a }^{ 2 } }  } $
$\displaystyle \Rightarrow \quad \frac { 16 }{ 25 } =1-\frac { { b }^{ 2 } }{ 25 } \Rightarrow { b }^{ 2 }=9$
Hence, equation of an ellipse is
$\displaystyle \frac { { x }^{ 2 } }{ 25 } +\frac { { y }^{ 2 } }{ 9 } =1\Rightarrow 9{ x }^{ 2 }+25{ y }^{ 2 }=225$

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)