Eccentricity of Hyperbola

Q. Let the foci of thhe ellipse $\frac{x^2}{16}+\frac{y^2}{7}=1$ and the hyperbola $\frac{x^2}{144}-\frac{y^2}{\alpha }=\frac{1}{25}$ coincide. Then the length of the latus rectum of the hyperbola is:

Q. $PQ$ is a double ordinate of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that $OPQ$ is an equilateral triangle, $O$ being the centre of the hyperbola, then the range of the eccentricity $e$ of the hyperbola is

1. $1<e<2$
2. $1<e<\sqrt{2}$
3. $e\ge \frac{2}{\sqrt{3}}$
4. $e<4$

Q. If $(5,12)$ and $(24,7)$ are the foci of a conic passing through the origin, then the eccentricity of conic can be:

1. $\frac{\sqrt{386}}{12}$
2. $\frac{\sqrt{386}}{38}$
3. $\frac{\sqrt{386}}{13}$
4. $\frac{\sqrt{386}}{25}$

Q. If $5x+9=0$ is the directrix of the hyperbola $16x^2-9y^2=144,$ then its corresponding focus is

1. $(-5,0)$
2. $(5,0)$
3. $(-\frac{5}{3},0)$
4. $(\frac{5}{3},0)$

Q. If the length of the transverse and conjugate axes of a hyperbola are $8$ and $6$ units respectively, then the absolute value of difference of focal distances of any point of the hyperbola is unit.

Q. The eccentricity of the hyperbola $\frac{x^2}{9}-\frac{y^2}{16}=1$ is

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

Q.

The eccentricity of the conic$9x^2-16y^2=144$ is

1. $\frac{5}{4}$

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

3. $\sqrt{7}$

4. $\frac{4}{3}$

Q.

A parabola is drawn with its focus at $(3,4)$and vertex at the focus of the parabola $y^2-12x-4y+4=0$. The equation of the parabola is

1. $y^2–8x–6y+25=0$

2. $y^2–6x+8y–25=0$

3. $x^2–6x–8y+25=0$

4. $x^2+6x–8y–25=0$

Q.

The degree and order of the differential equation of the family of all parabolas whose axis is $x$-axis are respectively

1. $2,1$

2. $1,2$

3. $3,2$

4. $2,3$

Q. The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is___ .

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

Q.

Two sources of sound $S_1$​ and $S_2$ produce sound waves of the same frequency $660Hz$. A listener is moving from source $S_1$​ towards $S_2$​ with a constant speed $um/s$ and he hears $10beats/s$. The velocity of sound is $330m/s$. Then, $u$ equals:

1. $5.5m/s$

2. $15m/s$

3. $2.5m/s$

4. $10m/s$

Q.

Find the eccentricity of the hyperbola, the length of whose conjugate axis is$\frac{3}{4}$ of the length of transverse asxis.

Q.

$\text{The eccentricity of the hyperbola}{x}^2-4y^2=1is$

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

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

3. $\frac{2}{\sqrt{5}}$

4. $\frac{\sqrt{5}}{2}$

Q. A hyperbola has its centre at the origin, passes through the point $(4,2)$ and has transverse axis of length $4$ unit along the $x-$axis. Then the eccentricity of the hyperbola is

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

Q. Let an ellipse and a hyperbola have same foci. If the length of conjugate axis of the hyperbola is equal to the length of minor axis of the ellipse, then the value of $\frac{1}{e_1^2}+\frac{1}{e_2^2}$ is $(e_1$ and $e_2$ denote the eccentricities of the two conics$)$

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

Q. The vertices of a hyperbola are at $(0,0)$ and $(10,0)$ and one of its foci is at $(18,0)$. The equation of the hyperbola is

1. $\frac{x^2}{25}-\frac{y^2}{144}=1$
2. $\frac{(x-5{)}^2}{25}-\frac{y^2}{144}=1$
3. $\frac{x^2}{25}-\frac{(y-5{)}^2}{144}=1$
4. $\frac{(x-5{)}^2}{25}-\frac{(y-5{)}^2}{144}=1$

Q. If a directrix of a hyperbola centred at the origin and passing through the point $(4,-2\sqrt{3})$ is $5x=4\sqrt{5}$ and its eccentricity is $e$, then :

1. $4e^4-12e^2-27=0$
2. $4e^4+8e^2-35=0$
3. $4e^4-24e^2+35=0$
4. $4e^4-24e^2+27=0$

Q. If the vertices of a hyperbola be at $(-2,0)$ and $(2,0)$ and one of its foci be at $(-3,0),$ then which one of the following points does not lie on this hyperbola

1. $(4,\sqrt{15})$
2. $(2\sqrt{6},5)$
3. $(-6,2\sqrt{10})$
4. $(6,5\sqrt{2})$

Q.

The eccentricity of the hyperbola whose latus-rectum is half of its transverse axis,

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

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

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

4. none of these

Q. Area bounded by the parabola $y=4x^2$, y-axis and the lines y=1, y=4 is

1. 3 sq. unit
2. $\frac{7}{5}$ sq. unit
3. $\frac{7}{3}$ sq. unit
4. None of these

Q. A hyperbola having transverse axis of length $\frac{1}{2}$ unit is confocal (having same foci) with the ellipse $3x^2+4y^2=12,$ then

1. equation of the hyperbola is $x^2-\frac{y^2}{15}=\frac{1}{16}$
2. eccenticity of the hyperbola is $4$
3. distance between the directrices of the hyperbola is $\frac{1}{8}$ unit
4. length of latus ractum of hyperbola is $\frac{15}{2}$ unit

Q. Let the curves $\frac{x^2}{a^2}+\frac{y^2}{b^2}=4$ and $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ have same foci. If the eccentricity of the given curves are $e_1$ and $e_2$ respectively, then

1. $e_2=\sqrt{\frac{32}{17}}$
2. $e_1=\sqrt{\frac{2}{17}}$
3. $e_2=\sqrt{\frac{8}{5}}$
4. $e_1=\sqrt{\frac{2}{5}}$

Q. If a variable line has its intercepts on the coordinates axes as $e,e^{\prime }$ where $\frac{e}{2},\frac{e^{\prime }}{2}$ are the eccentricities of a hyperbola and its conjugate hyperbola respectively, then the line always touches the circle centred at $O$ whose radius $r$ is equal to

Q. If the ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ and hyperbola $\frac{x^2}{144}-\frac{y^2}{81}=\frac{1}{25}$ intersect orthogonally, then the value of $b^2$ is

Q.

The length of the parabola $y^2=12x$ cut off by the latus rectum is

1. $6(\sqrt{2}+\log \left(\sqrt{2}+1\right))$

2. $3(\sqrt{2}+\log \left(\sqrt{2}+1\right))$

3. $6(\sqrt{2}-\log \left(\sqrt{2}+1\right))$

4. $3(\sqrt{2}-\log \left(\sqrt{2}+1\right))$

Q. For the hyperbola $16x^2-3y^2-32x+12y-44=0,$ the correct option(s) is/are

1. its length of transverse axis is $2\sqrt{3}$ units
2. its length of conjugate axis is $8$ units
3. its centre is $(1,2)$
4. its eccentricity is $\frac{\sqrt{19}}{3}$

Q. For the hyperbola $3y^2-16x^2-12y+32x-52=0$ the correct option(s) is/are

1. its length of transverse axis is $8$ units
2. its length of conjugate axis is $2\sqrt{3}$ units
3. end points of one latus rectum will be $(\frac{1}{4},2+\sqrt{19})$ and $(\frac{7}{4},2+\sqrt{19})$
4. Distance between the two directrices is $\frac{16}{\sqrt{19}}$ units

Q. If the centre, vertex and focus of a hyperbola are $(4,3),(8,3),(10,3)$ respectively, then the equation of the hyperbola is

1. $\frac{(x-4{)}^2}{16}-\frac{(y-3{)}^2}{24}=1$
2. $\frac{(x-4{)}^2}{16}-\frac{(y-3{)}^2}{20}=1$
3. $\frac{(x-4{)}^2}{20}-\frac{(y-3{)}^2}{16}=1$
4. $\frac{(x-4{)}^2}{20}-\frac{(y-3{)}^2}{12}=1$

Q. The angle between asymptotes of the standard hyperbola (centered at origin and $x$-axis as transverse axis) is $\frac{\pi }{3}$. If the radius of the director circle of the hyperbola is $4$ then the equation of the hyperbola is

1. $\frac{x^2}{3}-\frac{y^2}{1}=8$
2. $\frac{x^2}{25}-\frac{y^2}{9}=1$
3. $\frac{x^2}{2}-\frac{y^2}{1}=8$
4. $\frac{x^2}{25}-\frac{y^2}{16}=1$

Q. The locus of the point of intersection of the lines, $\sqrt{2}x-y+4\sqrt{2}k=0$ and $\sqrt{2}kx+ky-4\sqrt{2}=0$ ($k$ is any non-zero real parameter), is:

1. an ellipse whose eccentricity is $\frac{1}{\sqrt{3}}$.
2. a hyperbola with length of its transverse axis $8\sqrt{2}$
3. an ellipse with length of its major axis $8\sqrt{2}$
4. a hyperbola whose eccentricity is$\sqrt{3}$