Eccentric Angle : Ellipse

Q. If the chord through the points whose eccentric angles are $\theta$ and $\varphi$ on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ passes through a focus, then possible value(s) of $\tan \left(\frac{\theta }{2}\right)\tan \left(\frac{\varphi }{2}\right)$ is/are

1. $\frac{1}{9}$
2. $-9$
3. $-\frac{1}{9}$
4. $9$

Q. If $AOB$ is the positive quadrant of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ in which $OA=a,OB=b.$ Then area enclosed between acr $AB$ and chord $AB$ of the ellipse is

1. $\frac{ab}{2}(\pi -4)$ sq. unit
2. $ab(\pi -2)$ sq. unit
3. $\frac{ab}{4}(\pi -2)$ sq. unit
4. $\frac{ab}{4}(\pi -4)$ sq. unit

Q. If $AOB$ is the positive quadrant of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ in which $OA=a,OB=b.$ Then area enclosed between acr $AB$ and chord $AB$ of the ellipse is

1. $\frac{ab}{2}(\pi -4)$ sq. unit
2. $ab(\pi -2)$ sq. unit
3. $\frac{ab}{4}(\pi -2)$ sq. unit
4. $\frac{ab}{4}(\pi -4)$ sq. unit

Q.

Find the eccentric angle of a point on the ellipse $\frac{x^2}{6}+\frac{y^2}{2}=1$ whose distance from centre is 2.

1. ${45}^{\circ }$
2. ${30}^{\circ }$
3. ${135}^{\circ }$
4. ${75}^{\circ }$

Q. $S$ is one focus of an standard horizontal ellipse and $P$ is any point on the ellipse. If the maximum and minimum values of $SP$ are $m$ and $n$ respectively, then the length of semi minor axis is

1. $AM$ of $m,n$
2. $GM$ of $m,n$
3. $HM$ of $m,n$
4. none of these

Q. The eccentric angle of a point on the ellipse $\frac{x^2}{6}+\frac{y^2}{2}=1$ whose distance from the centre of the ellipse is $\sqrt{5},$ units is/are

1. $\frac{\pi }{6}$
2. $\frac{3\pi }{6}$
3. $\frac{5\pi }{6}$
4. $\frac{11\pi }{6}$

Q. If line $lx+my+n=0$ cuts the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at points whose eccentric angles differ by $\frac{\pi }{2}$, then the value of $\frac{a^2{l}^2+b^2{m}^2}{n^2}$ is

Q. The equation of chord of ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ whose sum and difference of eccentric angles are $\frac{\pi }{3}$ and $\frac{2\pi }{3}$ respectively is

1. $2\sqrt{3}x-3y=6$
2. $2\sqrt{3}x+3y=6$
3. $2\sqrt{3}x+3y=\sqrt{3}$
4. $2\sqrt{3}x+3y=6\sqrt{3}$

Q. The eccentric angle of a point on the ellipse $\frac{x^2}{6}+\frac{y^2}{2}=1$ whose distance from the centre of the ellipse is $\sqrt{5},$ units is/are

1. $\frac{\pi }{6}$
2. $\frac{3\pi }{6}$
3. $\frac{5\pi }{6}$
4. $\frac{11\pi }{6}$

Q. The eccentric angle of a point on the ellipse
$\frac{x^2}{6}+\frac{y^2}{2}=1$ whose distance from the center of the ellipse is $\sqrt{5}$ is:

1. $\frac{\pi }{6}$
2. $\frac{5\pi }{6}$
3. $\frac{7\pi }{6}$
4. $\frac{11\pi }{6}$

Q. The eccentric angle of point of intersection of the ellipse $x^2+4y^2=4$ and the parabola $x^2+1=y$ is

1. $\frac{\pi }{4}$
2. $\frac{\pi }{2}$
3. $\frac{\pi }{6}$
4. $\frac{\pi }{3}$