Definition of Hyperbola

Q. If $e_1$ and $e_2$ are the eccentricities of the ellipse, $\frac{x^2}{18}+\frac{y^2}{4}=1$ and the hyperbola, $\frac{x^2}{9}-\frac{y^2}{4}=1$ respectively and $(e_1,e_2)$ is a point on the ellipse, $15x^2+3y^2=k$. Then $k$ is equal to

1. $17$
2. $14$
3. $16$
4. $15$

Q. Let $\theta$ be the acute angle between the tangents to the ellipse $\frac{x^2}{9}+\frac{y^2}{1}=1$ and the circle $x^2+y^2=3$ at their point of intersection in the first quadrant. Then $\tan \theta$ is equal to :

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

Q.

Derive the equation of parabola $y^2=4ax$ in standard form.

Q. If the latus rectum subtends a right angle at the centre of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, then eccentricity $e$ is

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

Q.

If P is a point on the rectangular hyperbola $x^2-y^2=a^2,$ C is its centre and $S,S^{\prime }$ are the two foci, the $SP\cdot S^{\prime }P=$

1. 2

2. $(CP{)}^2$

3. $(CS{)}^2$

4. $(S{S}^{\prime }{)}^2$

Q.

If the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is reciprocal to that of the ellipse $x^2+4y^2=4$.

If the hyperbola passes through the focus of the ellipse, then

1. The equation of the hyperbola is $\frac{x^2}{3}-\frac{y^2}{2}=1$

2. A focus of the hyperbola is $\left(2,0\right)$

3. The eccentricity of the hyperbola is $\sqrt{\frac{5}{3}}$

4. The equation of the hyperbola is $x^2-3y^2=3$

Q.

The eccentricity of the hyperbola $x^2-y^2=25$ is

1. 2

2. 

3. 

4. 

Q. Let $H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,\text{where}a>b>0,$be a hyperbola in the $xy-$plane whose conjugate axis $LM$ subtends an angle of ${60}^{\circ }$ at one of its vertices $N.$ Let the area of the triangle $LMN$ be $4\sqrt{3}.$ sq. unit

$\begin{array}{cccc}& \text{LIST-I}& & \text{LIST-II}\\ P.& \text{The length of the conjugate axis of}H\text{is}& 1.& 8\\ Q.& \text{The eccentricity of}H\text{is}& 2.& \frac{4}{\sqrt{3}}\\ R.& \text{The distance between the foci of}H\text{is}& 3.& \frac{2}{\sqrt{3}}\\ S.& \text{The length of the latus rectum of}H\text{is}& 4.& 4\end{array}$

The correct option is

1. $P\to 4;Q\to 2;R\to 1;S\to 3$
2. $P\to 4;Q\to 3;R\to 1;S\to 2$
3. $P\to 4;Q\to 1;R\to 3;S\to 2$
4. $P\to 3;Q\to 4;R\to 2;S\to 1$

Q. If point $P(x,y)$ is such that it moves on a hyperbola $|\sqrt{(x-3{)}^2+(y-4{)}^2}-\sqrt{x^2+y^2}|=k^2+1$, then the number of possible integral value(s) of $k$ is

Q. The exhaustive set of values of $a^2$such that there exists a tangent to the ellipse$x^2+a^2{y}^2=a^2$such that the portion of the tangent intercepted by the hyperbola $a^2{x}^2-y^2=1$ subtends a right angle at the origin.

1. $[\frac{\sqrt{5}-1}{2},1)$
2. $[\frac{\sqrt{5}-1}{2},1)\cup (1,\frac{\sqrt{5}+1}{2}]$
3. $\left[\begin{array}{c}\frac{\sqrt{5}+1}{2},2\end{array}\right]$
4. $(1,2]$

Q.

The eccentricity of the hyperbola in the standard form $\left(\frac{x^2}{a^2}\right)+\left(\frac{y^2}{b^2}\right)=1$, passing through $\left(3,0\right)$ and $\left(3\sqrt{2},2\right)$ is

1. $\frac{13}{3}$

2. $\sqrt{13}$

3. $\sqrt{3}$

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

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

Q.

A hyperbola passing through a focus of the ellipse $\frac{x^2}{169}+\frac{y^2}{25}=1$.

Its transverse axis and conjugate axes coincide respectively with the major and minor axes of the ellipse.

The product of eccentricities is $1$. Then, the equation of the hyperbola is

1. $\frac{x^2}{144}-\frac{y^2}{9}=1$

2. $\frac{x^2}{169}-\frac{y^2}{25}=1$

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

4. $\frac{x^2}{25}-\frac{y^2}{9}=1$(x² / 25) – (y² / 9) = 1

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. $\frac{x^2}{9}–\frac{y^2}{16}=1$

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. Which of the following equation(s) ($t$ being the parameter) represents a hyperbola?

1. $\frac{tx}{a}-\frac{y}{b}+t=0,\frac{x}{a}+\frac{ty}{b}-1=0$
2. $x=\frac{a}{2}(t+\frac{1}{t}),y=\frac{b}{2}(t-\frac{1}{t})$
3. $x=e^t+e^{-t},y=e^t-e^{-t}$
4. $x^2=2(\cos t+3),y^2=2({\cos }^2\frac{t}{2}-1)$

Q. If two circles are given such that they neither intersect nor touches each other. Then the locus of centre of variable circle which touches both the circles externally is

1. circle
2. parabola
3. ellipse
4. hyperbola

Q.

The area of the plane region bounded by the curves $\mathrm{x}+2{\mathrm{y}}^2=0$ and $\mathrm{x}+3{\mathrm{y}}^2=1$ is equal to

1. $\frac{4}{3}\mathrm{sq}\mathrm{units}$

2. $\frac{5}{3}\mathrm{sq}\mathrm{units}$

3. $\frac{1}{3}\mathrm{sq}\mathrm{units}$

4. $\frac{2}{3}\mathrm{sq}\mathrm{units}$

Q. If the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and hyperbola $\frac{x^2}{a^2}-\frac{y^2}{4}=1$ intersect each other orthogonally, then the value of $a^2$ is

Q. The eccentricity of the hyperbola whose length of latus rectum is half the length of transverse axis is

1. $\sqrt{\frac{3}{2}}$
2. $2$
3. $\frac{8}{3}$
4. $\frac{3}{2}$

Q.

$\text{Find the ecentrictity, coordinates of foci, equations of directrices and length of the latus-rectum of the hyperbola}\left(i\right)9x^2-16y^2=144\left(ii\right)16x^2-9y^2=-144\left(iiii\right)4x^2-3y^2=36\left(iv\right)3x^2-y^2=4\left(v\right)2x^2-3y^2=5$

Q. The foot of the perpendicular from the focus to an asymptote of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}$= 1 is

1. 
2. 
3. 
4. 

Q. If $z\in C$ and ${log}_{|z+i|}|3+4i|<{log}_{|z+i|}|5+12i|$, then z lies

1. Inside the disc
2. Outside the curve z = - i +
3. Inside a circle passing through the origin
4. Outside a circle passing through the origin

Q. The equation of hyperbola whose axes are the coordinates axes, one vertex is $(-3,0)$ and eccentricity is $\sqrt{\frac{5}{2}}$ is

1. $\frac{x^2}{9}-\frac{2y^2}{27}=1$

2. $\frac{x^2}{9}-\frac{y^2}{27}=1$
3. $\frac{x^2}{15}-\frac{y^2}{26}=1$
4. $\frac{x^2}{9}-\frac{3y^2}{27}=1$

Q. If e and e’ are the eccentricities of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and its conjugate hyperbola the value of $\frac{1}{e^2}+\frac{1}{e^{‘2}}$ is

1. 2
2. 1
3. 3
4. 0

Q. Let $H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,\text{where}a>b>0,$be a hyperbola in the $xy-$plane whose conjugate axis $LM$ subtends an angle of ${60}^{\circ }$ at one of its vertices $N.$ Let the area of the triangle $LMN$ be $4\sqrt{3}.$ sq. unit

$\begin{array}{cccc}& \text{LIST-I}& & \text{LIST-II}\\ P.& \text{The length of the conjugate axis of}H\text{is}& 1.& 8\\ Q.& \text{The eccentricity of}H\text{is}& 2.& \frac{4}{\sqrt{3}}\\ R.& \text{The distance between the foci of}H\text{is}& 3.& \frac{2}{\sqrt{3}}\\ S.& \text{The length of the latus rectum of}H\text{is}& 4.& 4\end{array}$

The correct option is

1. $P\to 4;Q\to 2;R\to 1;S\to 3$
2. $P\to 4;Q\to 3;R\to 1;S\to 2$
3. $P\to 4;Q\to 1;R\to 3;S\to 2$
4. $P\to 3;Q\to 4;R\to 2;S\to 1$

Q.

If the length of the latus rectum and the length of the transverse axis of a hyperbola are $4\sqrt{3}$ and $2\sqrt{3}$ respectively, then the equation of the hyperbola is.

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

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

3. $\frac{x^2}{6}-\frac{y^2}{4}=1$

4. $\frac{x^2}{6}-\frac{y^2}{3}=1$

5. $\frac{x^2}{3}-\frac{y^2}{6}=1$

Q. If the latus rectum subtends a right angle at the centre of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, then eccentricity $e$ is

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

Q. Let the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ 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 at :

1. $(2,0)$
2. $(0,2)$
3. $(3,0)$
4. $(0,3)$

Q.

If $e_1$ and $e_2$ are the eccentricities of the ellipse, $\frac{x^2}{18}+\frac{y^2}{4}=1$ and the hyperbola, $\frac{x^2}{9}-\frac{y^2}{4}=1$ respectively and $(e_1,e_2)$ is a point on the ellipse, $15x^2+3y^2=k$. Then $k$ is equal to:

1. $14$

2. $15$

3. $17$

4. $16$

Q. If a hyperbola has length of its conjugate axis equal to $5$ unit and the distance between its foci is $13$ unit, then the eccentricity of the hyperbola is

1. $\frac{13}{6}$
2. $2$
3. $\frac{13}{8}$
4. $\frac{13}{12}$

Q. Tangents drawn from $(\alpha ,beta)$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ make angle ${\theta }_{1}and{\theta }_2$ with the x-axis.If $tan{\theta }_{1}tan{\theta }_2=1$, then $({\alpha }^2-bet{a}^2)$ =

1. 
2. 
3. 
4.