Eccentricity of Hyperbola

Q.

If the coordinates of two points $A,B$ are $\left(\sqrt{7},0\right)$ and $\left(-\sqrt{7},0\right)$ respectively and $P$ is any point on the conic, $9x^2+16y^2=144$, then $PA+PB$ is equal to

1. $6$

2. $16$

3. $9$

4. $8$

Q.

The length of the transverse axis of a hyperbola is $2\cos \alpha$.

The foci of the hyperbola are the same as that of the ellipse $9x^2+16y^2=144$. The equation of the hyperbola is

1. $\frac{x^2}{{\cos }^2\alpha }-\frac{y^2}{7-{\cos }^2\alpha }=1$

2. $\frac{x^2}{{\cos }^2\alpha }-\frac{y^2}{7+{\cos }^2\alpha }=1$

3. $\frac{x^2}{1+{\cos }^2\alpha }-\frac{y^2}{7-{\cos }^2\alpha }=1$

4. $\frac{x^2}{1+{\cos }^2\alpha }-\frac{y^2}{5-{\cos }^2\alpha }=1$

Q.

There are exactly two points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ whose distance from its center is the same and is equal to $\sqrt{\frac{a^2+2b^2}{2}}$​​. Then the eccentricity of the ellipse is

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

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

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

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

Q.

e and $e^1$ are the eccentricities of the hyperbolas $16x^2-9y^2=144$ and $9x^2-16y^2$= - 144 then e - $e^1$ =

1. 2

2. 1

3. 0

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

Q. The equations of the directrices of the hyperbola $16x^2-9y^2=-144$ are:

1. $y=\pm \frac{8}{25}$
2. $y=\pm \frac{16}{25}$
3. $y=\pm \frac{8}{5}$
4. $y=\pm \frac{16}{5}$

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 of the following is/are correct?

1. eccentricity of the hyperbola is $\frac{3}{2}$
2. $(6,5\sqrt{2})$ does not lie on hyperbola
3. length of the latus rectum is $5$ unit
4. $(2\sqrt{6},5)$ does not lie on hyperbola

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. An ellipse passes through the foci of the hyperbola, $9x^2–4y^2=36$ and its major and minor axis lie along the transverse and conjugate axis of the hyperbola respectively. If the product of eccentricities of the two conics is $\frac{1}{2},$ then which of the following points does not lie on the ellipse?

1. $(\sqrt{\frac{13}{2}},\sqrt{6})$
2. $(\sqrt{13},0)$
3. $(\frac{1}{2}\sqrt{13},\frac{\sqrt{3}}{2})$
4. $(\frac{\sqrt{39}}{2},\sqrt{3})$

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. 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. A bullet is fired and it hits a target. An observer in the same plane heard two sounds: the crack of the rifle and the thud of the bullet striking the target at the same instant. Then the locus of the observer is a

1. Ellipse
2. Hyperbola
3. Parabola
4. Circle

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

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

Q. If $e_1$ is the eccentricity of the ellipse $\frac{x^2}{16}+\frac{y^2}{25}=1$ and $e_2$ is the eccentricity of the hyperbola passing through the foci of the ellipse and $e_1.e_2=1$, then the equation of the hyperbola is

1. $\frac{16x^2}{81}-\frac{y^2}{9}=-1$
2. $\frac{16x^2}{81}-\frac{y^2}{9}=1$
3. $\frac{x^2}{9}-\frac{16y^2}{81}=-1$
4. $\frac{x^2}{9}-\frac{16y^2}{81}=1$

Q. The equation of the hyperbola $4x^2-32x-y^2-4y+24=0$ in its standard form is

1. $\frac{(x-4{)}^2}{16}$- \dfrac{(y+2)^2}{36}=1$
2. $\frac{(x-4{)}^2}{9}$- \dfrac{(y-2)^2}{36}=1$
3. $\frac{(x-4{)}^2}{9}$- \dfrac{(y+2)^2}{36}=1$
4. $\frac{(x-4{)}^2}{9}$- \dfrac{(y+2)^2}{16}=1$

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

Q. Equation of normal to the hyperbola $xy=c^2$ at the point $P(x_1,y_1)$ is $x{x}_1-y{y}_1={x_1}^2-{y_1}^2.$

1. True
2. False

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. 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 $5x+9=0$ 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 eccentricities of the hyperbolas $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1and\frac{y^2}{b^2}-\frac{x^2}{a^2}=1$ be e and $e_1$, then $\frac{1}{e^2}+\frac{1}{e_1^2}=$

1. 1

2. 2

3. 3

4. None of these