Eccentric Angle : Ellipse

Q. If $\alpha ,\beta$ are the eccentric angles of the extremities of a focal chord of an ellipse, then eccentricity of the ellipse is

1. $\frac{\sin \alpha -\sin \beta }{\sin (\alpha -\beta )}$
2. $\frac{\cos \alpha -\cos \beta }{\cos (\alpha -\beta )}$
3. $\frac{\cos \alpha +\cos \beta }{\cos (\alpha +\beta )}$
4. $\frac{\sin \alpha +\sin \beta }{\sin (\alpha +\beta )}$

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. Equation of the auxiliary circle of the ellipse
$\frac{x^2}{12}+\frac{y^2}{18}=1$ is

1. $x^2+y^2=9$
2. $x^2+y^2=18$
3. $x^2+y^2=12$
4. $x^2+y^2=30$

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

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

Q. The eccentric angle of a point on the ellipse $\frac{x^2}{6}+\frac{y^2}{2}=1$ whose distance from the 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 tangent at a point whose eccentric angle is ${60}^{\circ }$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b),$ meets the auxiliary circle at $L$ and $M$. If $LM$ subtends a right angle at the centre, then eccentricity of the ellipse is

1. $\frac{2}{\sqrt{7}}$
2. $\frac{1}{\sqrt{7}}$
3. $\frac{3}{\sqrt{7}}$
4. $\frac{1}{2}$

Q. Which of the following option(s) is/are correct for ellipse $4(x-2y+1{)}^2+9(2x+y+2{)}^2=25$

1. centre of ellipse is $(-1,0)$
2. eccentricity $=\frac{\sqrt{5}}{3}$
3. length of latus rectum is $\frac{4\sqrt{5}}{9}$ units
4. ends of minor axis are
   $(-\frac{1}{3},\frac{1}{3})$ and $(-\frac{5}{3},-\frac{1}{3})$

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 point of intersection of the ellipse $x^2+4y^2=4$ and the parabola $x^2+1=y$ is

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

Q.

Find the area of the region lying in the first quadrant and bounded by $y=4x^2,x=0,y=1$ and y = 4.

Q. The equation of auxiliary circle of the ellipse $16x^2+25y^2+32x-100y=284$ is

1. $x^2+y^2+2x-4y-20=0$
2. $x^2+y^2+2x-4y=0$
3. $(x+1{)}^2+(y-2{)}^2=400$
4. $(x+1{)}^2+(y-2{)}^2=225$

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 what is the area enclosed between arc $AB$ and chord $AB$ of the given ellipse?

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. Locus of the point which divides double ordinates of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$ in the ratio $1:2$ internally is

1. $\frac{x^2}{a^2}+\frac{9y^2}{b^2}=1$
2. $\frac{9x^2}{a^2}+\frac{9y^2}{b^2}=1$
3. $\frac{x^2}{a^2}+\frac{9y^2}{b^2}=\frac{1}{9}$
4. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=a^2-b^2$

Q.

The equation $12x^2+7xy+a{y}^2+13a-y+3=0$ represents a pair of perpendicular lines. Then, the value of $a$ is

1. $\frac{7}{2}$

2. $-19$

3. $-12$

4. $12$

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 value of $\sin \left(2{\sin }^{-1}\left(0.8\right)\right)$ is equal to

1. $\sin {1.2}^{\circ }$
2. $\sin {1.6}^{\circ }$
3. $0.48$
4. $0.96$

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. 
2. 
3. 
4. 

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

Q. Which of the following option(s) is/are correct for ellipse $4(x-2y+1{)}^2+9(2x+y+2{)}^2=25$

1. centre of ellipse is $(-1,0)$
2. eccentricity $=\frac{\sqrt{5}}{3}$
3. length of latus rectum is $\frac{4\sqrt{5}}{9}$ units
4. ends of minor axis are
   $(-\frac{1}{3},\frac{1}{3})$ and $(-\frac{5}{3},-\frac{1}{3})$

Q. $\text{If}f\left(x\right)=\underset{t\to \infty }{lim}\frac{(1+\cos \frac{\pi x}{2}{)}^t-1}{(1+\cos \frac{\pi x}{2}{)}^t+1},$ then which of the following is/are correct?

1. $\underset{x\to 1^{+}}{lim}f\left(x\right)=-1$
2. $\underset{x\to 1^{-}}{lim}f\left(x\right)=1$
3. $\underset{x\to 2^{+}}{lim}f\left(x\right)=1$
4. $\underset{x\to 2^{-}}{lim}f\left(x\right)=-1$

Q. If $\alpha$, $\beta$, $\gamma$ are the angles which a line makes with positive direction of the axes, then

Q. $\underset{h\to 0}{lim}\frac{{(a+h)}^2\sin (a+h)-a^2\sin a}{h}$ is $a(a\cos a+b\sin a).$

Q.

If the latus rectum of an ellipse be equal to half of its minor axis, then its eccentricity is

1. 3/2

2. 

3. 2/3

4. 

Q. If $\alpha ,\beta$ are the eccentric angles of the extremities of a focal chord of an ellipse, then eccentricity of the ellipse is

1. $\frac{\cos \alpha +\cos \beta }{\cos (\alpha +\beta )}$
2. $\frac{\sin \alpha -\sin \beta }{\sin (\alpha -\beta )}$
3. $\frac{\cos \alpha -\cos \beta }{\cos (\alpha -\beta )}$
4. $\frac{\sin \alpha +\sin \beta }{\sin (\alpha +\beta )}$

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 value of sin ${18}^0$

1. $\frac{\sqrt{5}-1}{4}$
2. $\frac{\sqrt{5}+1}{4}$
3. $-\frac{\sqrt{5}+1}{4}$
4. $\frac{1-\sqrt{5}}{4}$

Q. If $\alpha ,\beta$ are the eccentric angles of the extremities of a focal chord of an ellipse, then eccentricity of the ellipse is

Q. $f:R\to R,g:R\to R$ and $f\left(x\right)=\sin x$, $g\left(x\right)=x^2$ then $fog\left(x\right)=$

1. $x^2\sin x$
2. $x^2+\sin x$
3. ${\sin }^{2}x$
4. $\sin x^2$

Q. Let $f\left(x\right)=\{\begin{array}{c}\sin x,x\ge 0\\ -\sin x,x<0\end{array}$.

1. continuous at $x=0$
2. differentiable at $x=0$
3. discontinuous at $x=0$
4. not differentiable at $x=0$

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. $9$