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 equation of the latus rectum of the ellipse $9{x}^{2}+4{y}^{2}-18x-8y-23=0$ are

  1. $y=\pm \sqrt{5}$
  2. $y=- \sqrt{5}$
  3. $y=1\pm \sqrt{5}$
  4. $y=-1\pm \sqrt{5}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$9x^2+4y^2-18x-8y-23=0$
$=(3x-3)^2+(2y-2)^2-13-23=0$
$\Rightarrow \ 9(x-1)^2+4(y-1)^2=36$
$\Rightarrow \ \dfrac {(x-1)^2}{4}+\dfrac {(y-1)^2}{9}=1$
Shifting origin to $(1,1)\Rightarrow x-1=x,\ y-1=y$
$\Rightarrow \ \dfrac {x^2}{4}+\dfrac {y^2}{9}=1$
$a=2,= b=3,\ e^2=1+\dfrac {a^2}{b^2}=1+\dfrac {4}{9}=\dfrac {5}{9}$
$\Rightarrow \ e=\pm \sqrt {\dfrac {5}{9}}+\dfrac {\sqrt 5}{3}$
$\Rightarrow \ $ Focus $=(0,\ \pm be)=(0,\ \pm \sqrt 5)$
$\Rightarrow \ $ latus ractum $\Rightarrow \ y=\pm \sqrt {5}$
Shifting back, $y=y-1$
$\Rightarrow \ y-1=\pm \sqrt {5}$
$\Rightarrow \ y=1\pm \sqrt 5\  \Rightarrow \ (C) $


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

If there is exactly one tangent at a distance of $4$ units from one of the locus of $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{a^{2}-16}=1, a>4$, then length of latus rectum is :-

  1. $16$
  2. $\dfrac{8}{3}$
  3. $12$
  4. $15$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation represents a hyperbola for a > 4. The condition of exactly one tangent at a distance of 4 units implies the distance from the center to the tangent is 4, which corresponds to the distance to the asymptotes or a specific property of the hyperbola. Solving for the latus rectum length 2(a^2-16)/a leads to 16.

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. $r\in \left(2,\:\dfrac{7}{2}\right)\cup \left(\dfrac{7}{2},5\right)$
  3. $r>5$
  4. $r<2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Equating the equation of the ellipse with the second-degree equation
$A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0$ with $\dfrac{{x}^{2}}{2-r}+\dfrac{{y}^{2}}{r-5}+1=0$
we get $A=\dfrac{1}{2-r}, B=0, C=\dfrac{1}{r-5},D=0,E=0$ and $F=1$
For the second degree equation to represent an ellipse, the coefficients must satisfy the discriminant condition ${B}^{2}-4AC<0$ and also $A\neq C$
$\Rightarrow {\left(0\right)}^{2}-4\left(\dfrac{1}{2-r}\right)\left(\dfrac{1}{r-5}\right)<0$
$\Rightarrow -4\left(\dfrac{1}{2-r}\right)\left(\dfrac{1}{r-5}\right)<0$
$\Rightarrow \left(\dfrac{1}{2-r}\right)\left(\dfrac{1}{r-5}\right)>0$
$\Rightarrow \left(2-r\right)\left(r-5\right)<0$
$\Rightarrow \left(r-2\right)\left(r-5\right)>0$
$\Rightarrow r>2$
Multiple choice mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

Equation of the ellipse whose minor axis is equal to the distance between foci and whose latus rectum is $10 ,$ is given by ____________.

  1. $2 x ^ { 2 } + 3 y ^ { 2 } = 100$
  2. $2 x ^ { 2 } + 3 y ^ { 2 } = 80$
  3. $x ^ { 2 } + 2 y ^ { 2 } = 100$

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the ellipse be x^2/a^2 + y^2/b^2 = 1. Minor axis 2b = distance between foci 2ae, so b = ae. Latus rectum 2b^2/a = 10. Using b^2 = a^2(1-e^2) = a^2 - a^2e^2 = a^2 - b^2, we get a^2 = 2b^2. Substituting into the latus rectum formula: 2b^2/sqrt(2b^2) = 10, so sqrt(2)b = 10, b^2 = 50, a^2 = 100. The equation is x^2/100 + y^2/50 = 1, which multiplies to x^2 + 2y^2 = 100.

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

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

  1. centre is $(-1,2) $
  2. Length of axes are $ {\sqrt {3}} and 1 $
  3. eceentricity is $ \sqrt {\cfrac {2} {3}} $

  4. All of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given,

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

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

$\dfrac{(x+1)^2}{\frac{1}{4}}+\dfrac{(y-2)^2}{\frac{3}{4}}=1$

$\therefore a=\dfrac{1}{2},b=\dfrac{\sqrt 3}{2}$

⇒ Centre $ = (-1, 2)$

Here $b^2>a^2$

⇒ eccentricity$(e) = \sqrt {\dfrac {b^2-a^2}{b^2}} $

$= \sqrt {\dfrac {\dfrac 3 4 - \dfrac 1 4}{\dfrac 3 4}}=\sqrt{\dfrac 2 3}$

Length of arcs,

length of major arc $=2b=2\left ( \dfrac{\sqrt 3}{2} \right )=\sqrt 3$

length of minor arc $=2a=2\left ( \dfrac{1}{2} \right )=1$

Option D is correct.
Multiple choice mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

A point $P$ on the ellipse $\displaystyle \frac{x^{2}}{25} + \frac{y^{2}}{9} = 1$ has the eccentric angle $\displaystyle \frac{\pi}{8}$. The sum of the distance of $P$ from the two foci is

  1. $5$
  2. $6$
  3. $10$
  4. $3$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given,ellipse equation as $\dfrac{x^2}{25}+\dfrac{y^2}{9}=1$
Length of major axis, $a=5$ and length of minor axis, $b=3$
$P$ is a point on the ellipse whose eccentricity is $\dfrac{\pi}{8}.$
We know that, sum of the distances of any point on the ellipse from its foci equal to twice the major axis.
Let $S,S'$ be foci of ellipse and $a,b$ as the length of major,minor axis respectively.
$\Rightarrow SP+S'P=2a$
$\Rightarrow SP+S'P=2 \times 5=10$

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

Axes are coordinates axes, the ellipse passes through the points where the straight line $\dfrac {x}{4}+\dfrac {y}{3}=1$  meets the coordinates axes. Then equation of the ellipses is 

  1. $\dfrac {x^{2}}{16}+\dfrac {y^{2}}{9}=1$
  2. $\dfrac {x^{2}}{64}+\dfrac {y^{2}}{36}=1$
  3. $\dfrac {x^{2}}{4}+\dfrac {y^{2}}{3}=1$
  4. $\dfrac {x^{2}}{8}+\dfrac {y^{2}}{6}=1$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The line x/4 + y/3 = 1 meets the x-axis at (4, 0) and the y-axis at (0, 3). An ellipse centered at the origin passing through these points has the form x^2/a^2 + y^2/b^2 = 1. Substituting the points gives a^2 = 16 and b^2 = 9.