Tag: ellipse

Questions Related to ellipse

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

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

  1. $y=\pm \sqrt{5}$

  2. $y=- \sqrt{5}$

  3. $y=1\pm \sqrt{5}$

  4. $y=-1\pm \sqrt{5}$

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Correct Option: C
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) $


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 :-

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

  1. $16$

  2. $\dfrac{8}{3}$

  3. $12$

  4. $15$

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Correct Option: A

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

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

  1. $r>2$

  2. $r\in \left(2,:\dfrac{7}{2}\right)\cup \left(\dfrac{7}{2},5\right)$

  3. $r>5$

  4. $r<2$

Show answer
Correct Option: A
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 $A\neq C$
$\Rightarrow {\left(0\right)}^{2}-4\left(\dfrac{1}{2-r}\right)\left(\dfrac{1}{r-5}\right)<0$
$\Rightarrow -4\left(\dfrac{1}{2-r}\right)\left(\dfrac{1}{r-5}\right)<0$
$\Rightarrow \left(\dfrac{1}{2-r}\right)\left(\dfrac{1}{r-5}\right)>0$
$\Rightarrow \left(2-r\right)\left(r-5\right)<0$
$\Rightarrow \left(r-2\right)\left(r-5\right)>0$
$\Rightarrow r>2$

Distance between the foci of the curve represented by the equation $x=3+4\cos\theta, y=2+3\sin\theta$, is?

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

  1. $3\sqrt{7}$

  2. $2\sqrt{7}$

  3. $\sqrt{7}$

  4. $\dfrac{\sqrt{7}}{2}$

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Correct Option: A

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

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

  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

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Correct Option: C

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

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

  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

Show answer
Correct Option: B
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.

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

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

  1. $5$

  2. $6$

  3. $10$

  4. $3$

Show answer
Correct Option: C
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.
$\Rightarrow SP+S'P=2a$
$\Rightarrow SP+S'P=2 \times 5=10$

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 

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

  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. $\dfrac {x^{2}}{8}+\dfrac {y^{2}}{6}=1$

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Correct Option: A

The equation $\sqrt{(x-3)^{2}+(y-1)^{2}}+\sqrt{(x-3)^{2}+(y-1)^{2}}=6$ represents : 

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

  1. an ellipse

  2. a pair of straight lines

  3. a circle

  4. the line segment joining the point $(-3,1)$ to the point $(3,1)$

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Correct Option: A

If a chord of $y^{ 2 } = 4ax$ makes an angle $\alpha ,\alpha \epsilon \left( 0,\pi /4 \right)$ with the positive direction of $X-axis$, then the minimum length of this focal chord is 

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

  1. $2 \sqrt{ 2 } a units$

  2. $4 \sqrt{ 2 } a units$

  3. $8a units$

  4. $16 a units$

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Correct Option: A