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

Which of the following can be the equation of an ellipse?

  1. $x^{2} + y^{2} = 5$
  2. $\dfrac {x^{2}}{9} + \dfrac {x^{2}}{9} = 1$
  3. $2x^{2} + 3y^{2} = 5$
  4. $2x + 2y = 5$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

An ellipse equation in standard form is x^2/a^2 + y^2/b^2 = 1. Option C, 2x^2 + 3y^2 = 5, can be rewritten as x^2/(5/2) + y^2/(5/3) = 1, which fits the form of an ellipse.

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. $r > 2$
  2. $2 < r < 5$
  3. $r > 5$
  4. $r \in (2,5)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given $\dfrac{x^2}{2-r}+\dfrac{y^2}{r-5}+1=0$ represents a ellipse

$\implies \dfrac{x^2}{2-r}+\dfrac{y^2}{r-5}=-1$
$\implies \dfrac{x^2}{r-2}+\dfrac{y^2}{5-r}=1$
Since this equation is an ellipse so $r-2>0,5-r>0\implies 2<r<5$

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

The locus of center of a variable circle touching the circle of radius ${ r } _{ 1 }and{ r } _{ 2 }$ extemally which also touch each other externally , is a conic of the eccentricity $e$.If $\dfrac { { r } _{ 1 } }{ { r } _{ 2 } } =3+2\sqrt { 2 } $ then ${ e }^{ 2 }$ is 

  1. 2

  2. 3

  3. 4

  4. 5
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The locus of the center of a circle touching two circles of radii r1 and r2 externally is an ellipse with foci at the centers of the two circles. The distance between foci is 2ae = r1 + r2, and the major axis 2a = r1 + r2. Wait, the distance between centers is r1 + r2. The locus is a hyperbola if they touch externally. Given the context of eccentricity, the calculation leads to e^2 = 2.

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

The arrangement of the following conics in the descending order of their lengths of semi latus rectum is
A) $ 6= r (1 + 3\cos \theta )$
B) $10= r (1 + 3\cos \theta )$
C) $8= r (1 + 3\cos \theta )$
D) $12= r (1 + 3\cos \theta )$

  1. $D, A, B, C$
  2. $B, C, D, A$
  3. $D, B, C, A$
  4. $A, C, B, D$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Comparing given equation with standard equation $r(1+e\cos\theta)=l$ where $l$ is semi latus rectum
Hence order is $D,B,C,A$

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

The focal chord of a conic perpendicular to axis is 

  1. Tangent

  2. Vertex

  3. Focal distance

  4. Latus rectum
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A perpendicular from a point on the conic to the axis is called an ordinate, and if produced to meet the conic again it is called a double ordinate. The double ordinate through the focus is called the $latus\ rectum$.

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

The sum of the focal distances of a point on the ellipse $\cfrac { { x }^{ 2 } }{ 4 } +\cfrac { { y }^{ 2 } }{ 9 } =1$ is:

  1. $4$ units
  2. $6$ units
  3. $8$ units
  4. $10$ units
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The sum of focal distances from a point of ellipse is 2 times the major axis.
For the given ellipse , length of semi major axis i.e. $b$ is $3$.
So required length $=2\times 3=6$
Option B is true