Tag: ellipse

Questions Related to ellipse

Let $(a, 0)$ and $B(b, 0)$ be fixed distinct points on the $x-axis$, none of which coincides with the origin $O(0, 0)$ and let $C$ be a point on the $y-axis$. Let $L$ be a line through the $O(0, 0)$ and perpendicular to the line $AC$, The locus of the point of intersection of lines $L$ and $BC$ if $C$ varies along the $y-axis$, is (provided $x^{2}+ab\neq 0$) 

position of point wrt ellipse ellipse maths

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

  2. $\dfrac{x^{2}}{a}+\dfrac{y^{2}}{b}=y$

  3. $\dfrac{x^{2}}{b}+\dfrac{y^{2}}{a}=x$

  4. $\dfrac{x^{2}}{b}+\dfrac{y^{2}}{a}=y$

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Correct Option: D

If P($\theta$) and Q($\pi$/2 + $\theta$) are two points on the ellipse $\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Locus of the mid-point of PQ is

position of point wrt ellipse ellipse maths

  1. $\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{1}{2}$

  2. $\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 4$

  3. $\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 2$

  4. None of these

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Correct Option: A
Explanation:

Given,$P(\theta),Q(\frac{\pi}{2}+\theta)$ are two points on the ellipse $\displaystyle\frac{x^2}{a^2}+\displaystyle\frac{y^2}{b^2}=1$
Any point on the ellipse will be $(acos\theta,bsin\theta)$
$\Rightarrow P=(acos\theta,bsin\theta),Q=(acos(\frac{\pi}{2}+\theta),bsin(\frac{\pi}{2}+\theta))$
$\Rightarrow P=(acos\theta,bsin\theta),Q=(-asin\theta,bcos\theta)$
Let required point be $C(x,y)$
Given, $C=mid-point\;of\;PQ$
$\Rightarrow (x,y)=(\displaystyle\frac{(acos\theta-asin\theta)}{2},\displaystyle\frac{(bsin\theta+bcos\theta)}{2})$
$\Rightarrow \displaystyle\frac{x}{a}=(\displaystyle\frac{(cos\theta-sin\theta)}{2}),\displaystyle\frac{y}{b}=(\displaystyle\frac{(sin\theta+cos\theta)}{2})$
on squaring $\displaystyle\frac{x}{a},\displaystyle\frac{y}{b}$ and adding both
$\Rightarrow \displaystyle\frac{x^2}{a^2}+\displaystyle\frac{y^2}{b^2}=(\displaystyle\frac{(cos\theta-sin\theta)}{2})^2+(\displaystyle\frac{(sin\theta+cos\theta)}{2})^2$
$\Rightarrow \displaystyle\frac{x^2}{a^2}+\displaystyle\frac{y^2}{b^2}=\displaystyle\frac{2(cos^2\theta+sin^2\theta)}{4}$
$\Rightarrow \displaystyle\frac{x^2}{a^2}+\displaystyle\frac{y^2}{b^2}=\displaystyle\frac{1}{2}$(since $cos^2\theta+sin^2\theta=1$)

The value of $\alpha$ for which the point $(\alpha,\alpha+2)$ is an interior point of smaller segment of the curve $x^{2}+y^{2}-4=0$ made by the chord of the curve whose equation is $3x+4y+12=0$ is

position of point wrt ellipse ellipse maths

  1. $\left(-\infty,\dfrac {-20}{7}\right)$

  2. $(-2,0)$

  3. $\left(-\infty,\dfrac {20}{7}\right)$

  4. $\alpha\ \epsilon\ \phi$

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Correct Option: A

The distance of a point on the ellipse $\dfrac {x^{2}}{6}+\dfrac {y^{2}}{2}=1$ from the centre is $2$, then the eccentric angle is-

position of point wrt ellipse ellipse maths

  1. $\dfrac \pi3$

  2. $\dfrac \pi4$

  3. $\dfrac \pi6$

  4. $\dfrac \pi2$

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Correct Option: B
Explanation:
Given ellipse $ \dfrac{x^{2}}{6}+\dfrac{y^{2}}{2} = 1 $

Let $ \theta$ be the eccentric angle of the point $p$

coordinate of $p$ $ (\sqrt{6}cos\theta ,\sqrt{2}sin\theta )$

Given distance $= 2 $

$ \therefore $ $OP = 2$

$ \sqrt{6 cos^{2}\theta +2sin^{2}\theta } = 2 \Rightarrow 6cos^{2}\theta +2sin^{2}\theta  = 4$

$ 3cos^{2}\theta +sin^{2}\theta  = 2 $

$ 2 sin^{2}\theta  = 1$

$ sin^{2}\theta  = \dfrac{1}{2} \Rightarrow  sin\theta  = \pm  \dfrac{1}{\sqrt{2}}$

$ \therefore $ eccentric angle $\theta  = \pm \dfrac{\pi }{4}$

A rod of length $l$ rests against a vertical wall and a floor of a room.Let P be a point on the rod,nearer to its end on the wall, that divides its length in the ratio 1:2 if the rod begins to slide on the floor,then the locus of P is:

position of point wrt ellipse ellipse maths

  1. an ellipse of eccentricity $\dfrac { 1 }{ 2 }$

  2. an ellipse of eccentricity $\dfrac { \sqrt { 3 } }{ 2 }$

  3. a circle of radius $\dfrac { l }{ 2 }$

  4. a circle of radius $\dfrac { \sqrt { 3 } }{ 2 } l$

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Correct Option: B

The distance from the foci of $P(a,b)$ on the ellipse $\dfrac {x^{2}}{9}+\dfrac {y^{2}}{25}=1$ are

position of point wrt ellipse ellipse maths

  1. $4\pm \dfrac {5}{4}b$

  2. $5\pm \dfrac {4}{5}a$

  3. $5\pm \dfrac {4}{5}b$

  4. $none\ of\ these$

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Correct Option: C

The number of rational points on the ellipse $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{4}=1$ is

position of point wrt ellipse ellipse maths

  1. $\infty$

  2. $4$

  3. $0$

  4. $2$

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Correct Option: A

In an ellipse the distance between its foci is 6 and its minor axis is 8 . Its eccentricity is

position of point wrt ellipse ellipse maths

  1. $\dfrac{6}{5}$

  2. $\dfrac{4}{5}$

  3. $\dfrac{3}{5}$

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

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Correct Option: C
Explanation:
Given that $2ae=6$ and $2b=8$

$\Rightarrow ae=3$      $\Rightarrow b=4$

$b^2=a^2(1-e^2)$

$b^2=a^2-a^2e^2$

$16=a^2-9$

$\Rightarrow a^2=25$

$\Rightarrow a=5$

$5e=3$

$\Rightarrow e=\dfrac{3}{5}$.

A point on the ellipse is $\displaystyle \frac{x^{2}}{6} + \frac{y^{2}}{2} = 1$ at a distance of $2$ from the centre of the ellipse has the eccentric angle

position of point wrt ellipse ellipse maths

  1. $\displaystyle \frac{\pi}{4}$

  2. $\displaystyle \frac{\pi}{3}$

  3. $\displaystyle \frac{\pi}{6}$

  4. $\displaystyle \frac{\pi}{2}$

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Correct Option: A
Explanation:

Given, equation of ellipse as $\displaystyle\frac{x^2}{6}+\displaystyle\frac{y^2}{2}=1$(where length of major axis=$\sqrt6$,length of minor axis=$\sqrt2$)
center of ellipse is (0,0) which is parallel to horizontal axis and with eccentricity 'e'.
Any point on the ellipse will be as $(acos\theta,bsin\theta)\Rightarrow P(\sqrt6cos\theta,\sqrt2sin\theta)$
Distance of point P from center=2
$\Rightarrow \sqrt((\sqrt6cos\theta-0)^2+(\sqrt2sin\theta-0)^2)=2$
$\Rightarrow (6cos^2\theta+2sin^2\theta)=4$
$\Rightarrow (3cos^2\theta+sin^2\theta)=2$
$\Rightarrow 2cos^2\theta+1=2$
$\Rightarrow cos\theta=\pm\frac{1}{\sqrt2}$
$\Rightarrow \theta=\displaystyle\frac{\pi}{4}\;or\;\displaystyle\frac{-\pi}{4}$
Option $A$ is correct

The position of the point $(1, 3)$ with respect to the ellipse $4x^2+9y^2-16x-54y+61=0$.

position of point wrt ellipse ellipse maths

  1. Outside the ellipse

  2. On the ellipse

  3. On the major axis

  4. On the minor axis

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Correct Option: A