Eccentricity of Hyperbola

Q.

The locus of the vertices of the family of parabolas $y=\frac{a^3{x}^2}{3}+\frac{a^{2}x}{2}-2a$ is

1. $xy=\frac{3}{4}$

2. $xy=\frac{35}{16}$

3. $xy=\frac{64}{105}$

4. $xy=\frac{105}{64}$

Q.

Parametric equation of rectangular hyperbola $xy=c^2$ is $x=ct\&y=\frac{c}{t}$ where $tϵR$.

1. False

2. True

Q.

The shortest distance from the point $(1,2,-1)$ to the surface of the sphere $x^2+y^2+z^2=54$ is

1. $3\sqrt{6}$

2. $2\sqrt{6}$

3. $\sqrt{6}$

4. $2$

Q.

Find the equation of the hyperbola whose vertices are $(\pm 7,0)$ and the eccentricity is $\frac{4}{3}.$

Q.

The …… of a conic is the chord passing through the focus and perpendicular to the axis.

1. Line of axis

2. Directrix

3. Y-axis

4. Latus-rectum

Q.

The coordinates of the focus of the parabola described parametrically by $x=5t^2+2,y=10t+4$ are:

1. $(7,4)$

2. $(3,4)$

3. $(3,-4)$

4. $(-7,4)$

Q.

Find the coordinates of the foci, the vertices, the eccentricity and the length of the latus rectum of the hyperbola,

$5y^2-9x^2=36$

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 equation of the hyperbola is

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

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

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

4. None of these

Q. The latus rectum of a parabola whose focal chord is PSQ such that SP=3 and SQ=2, is given by

Q. 35. Length o latus rectum of the hyperbola xy-3x-4y+8=0 is ?

Q.

Let P(6, 3) be a point on hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$ If the normal at the point P intersect the x - axis at (9, 0), then the eccentricity of the hyperbola is

(IIT JEE 2011)

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

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

3. $\sqrt{3}$

4. $\sqrt{2}$

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. Write the eccentricity of the hyperbola 9x² − 16y² = 144.

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. Find the centre, eccentricity, foci and directrices of the hyperbola
(i) 16x² − 9y² + 32x + 36y − 164 = 0
(ii) x² − y² + 4x = 0
(iii) x² − 3y² − 2x = 8.

Q.

$e_1$ and $e_2$ are the eccentricities of two conics S and $S^1$. If ${e_1}^2+{e_2}^2$ = 3 then both S and $S^1$ can be

1. ellipse

2. parabolas

3. hyperbolas

4. circles

Q. The eccentricity of the conic $4(2y-x-3{)}^2-9(2x+y-1{)}^2=80$ is

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

Q. Let the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,(a>b)$ be the reciprocal to that of the ellipse $x^2+4y^2=4$. If the hyperbola passes through the focus of the ellipse, then the focus of the hyperbola is

1. $(\pm 2,0)$
2. $(\pm 2\sqrt{3},0)$
3. $(\pm \frac{2}{\sqrt{3}},0)$
4. $(\pm \frac{2}{3},0)$

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