Minor Axis of Ellipse

Q. If the minor axis of an ellipse subtends an angle 60° at each focus then the eccentricity of the ellipse is

1. 
2. 
3. none
4. 

Q.

If the major axis is "n” times the minor axis of the ellipse, then its eccentricity is

1. 

2. 

3. 

4. 

Q. A point $P$ moves inside a square of area $4$ sq.units such that it is nearer to point of intersection of its diagonal than any vertex. Area of region traced by $P$ is-

1. $4$
2. $2$
3. $3$
4. $1$

Q. The distance between the foci is equal to the minor axis of an ellipse then its eccentricity is

1. none
2. 
3. 
4. 

Q.

$Ifthefocioftheellipse\frac{x^2}{16}+\frac{y^2}{b^2}=1andthehyperbola\frac{x^2}{144}-\frac{y^2}{81}=\frac{1}{25}coincide,writethevalueof{b}^2$

Q.

If $e_1$ and $e_2$ are the eccentricities of a hyperbola$3x^2-3y^2=25$ and its conjugate, then

1. $e_1^2+e_2^2=2$

2. $e_1^2+e_2^2=4$

3. $e_1+e_2=4$

4. $e_1+e_2=\sqrt{2}$

Q. If the latus rectum of a hyperbola through one focus subtends 60° at the other focus then its eccentricity e =

1. 
2. 
3. 
4. 

Q. If the major axis of an ellipse is $3$ times its minor axis, then its eccentricity of the ellipse is

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

Q. If the major axis is “n” (n>1) times the minor axis of the ellipse, then its eccentricity is

1. 
2. 
3. 
4. 

Q. The ellipse $E_1:\frac{x^2}{9}+\frac{y^2}{4}=1$ is inscribed in a rectangle R whose sides are parallel to the coordinate axes, Another ellipse $E_2$ passing through the point (0, 4) circumscribes the rectangle R.The eccentricity of the ellipse $E_2$ is

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

Q. Let $A_1,G_1,H_1$ denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For $n\ge 2,$ let $A_{n-1}$ and $H_{n-1}$ hav arithmetic, gemetric and hrmonic means as $A_n,G_n,H_n$ respectively.

1. $G_1<G_2<G_3<...$
2. $G_1>G_2>G_3>...$
3. $G_1<G_3<G_5<...$ and $G_2>G_4>G_6>...$
4. $G_1=G_2=G_3=...$

Q. In each of the Exercises $1$ to $9$, find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
$\frac{x^2}{16}+\frac{y^2}{9}=1$

Q.

If the length of latusrectum of an Ellipse is equal to semi minor axis then Its eccentricity is

1. 

2. 

3. 

4. 

Q. If $f\left(x\right)$ is continuous in $[0,1]$ and $f\left(\frac{1}{3}\right)=1$ then $\underset{n\to \infty }{lim}f\left(\frac{n}{\sqrt{9n^2+1}}\right)$ is

1. $1$
2. $\frac{1}{3}$
3. none of these
4. $0$

Q. Assertion :If $P_1{Q}_1$ & $P_2{Q}_2$ are two focal chords of the parabola $y^2=4ax$, then locus of point of intersection of chords $P_1{P}_2$ & $Q_1{Q}_2$ is directrix of the parabola. Reason: The locus of points of intersection of perpendicular tangents of a parabola is the directrix of parabola.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. The extremities of a diagonal of a rectangular parallelopiped whose faces are parallel to the reference planes are $(-2,4,6)$ and $(3,16,6)$. The length of the base diagonal is

1. $13$
2. $\sqrt{13}$
3. $2\sqrt{13}$
4. $169$

Q.

If the major axis is "n” times the minor axis of the ellipse, then its eccentricity is

1. $\frac{\sqrt{n-1}}{n}$

2. $\frac{\sqrt{n-1}}{n^2}$

3. $\frac{\sqrt{n^2-1}}{n^2}$

4. $\frac{\sqrt{n^2-1}}{n}$

Q. The minor axis of the ellipse $9x^2+5y^2=30y$ is

1. $6$
2. $2\sqrt{5}$
3. $\sqrt{6}$
4. $2\sqrt{6}$

Q. $S$ is one focus of an 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 major axis is

1. AM of $m,n$
2. G.M. of $m,n$
3. HM of $m,n$
4. AGP of $m,n$

Q. Given $a^2+b^2+c^2=25;x^2+y^2+z^2=36\text{and}ax+by+cz=30$ for a, b, c being real. How will you prove $\frac{a}{x}=\frac{b}{y}=\frac{c}{z}=\frac{5}{6}$?