Directrix of Hyperbola

Q. The distance between the foci of a hyperbola is double the distance between its vertices and the length of its conjugate axis is $6$. Then which of the following is/are correct?

1. length of transverse axis $=\sqrt{3}$ unit
2. length of latus rectum $=3\sqrt{3}$ unit
3. length of transverse axis $=2\sqrt{3}$ unit
4. length of latus rectum $=6\sqrt{3}$ unit

Q. Equation of the locus of all points such that the difference of its distances from $(-3,-7)$ and $(-3,3)$ is $8$ units is

1. $\frac{(x+3{)}^2}{16}-\frac{(y+2{)}^2}{9}=1$
2. $\frac{(x+3{)}^2}{9}-\frac{(y+2{)}^2}{16}=-1$
3. $\frac{(x+3{)}^2}{7}-\frac{(y+2{)}^2}{19}=-1$
4. $\frac{(x+3{)}^2}{9}-\frac{(y+2{)}^2}{19}=1$

Q. The equation of transverse and conjugate axis of hyperbola are $x+2y-3=0,2x-y+4=0$ respectively, and their respective lengths are $\sqrt{2}$ and $\frac{2}{\sqrt{3}}$. The equation of hyberbola is

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

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

Q.

An ellipse has $OB$ as semi-minor axis $F$and $F$ its foci and the angle$FBF$ is a right angle. Then, the eccentricity of the ellipse is

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

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

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

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

Q.

A hyperbola, having the transverse axis of length $2\sin \theta$, is confocal with the ellipse $3x^2+y^2=12$ then, its equation is

1. $x^2\cos e{c}^2\theta –y^{2}se{c}^2\theta =1$

2. $x^{2}se{c}^2\theta –y^2\cos e{c}^2\theta =1$

3. $x^2{\sin }^2\theta –y^2{\cos }^2\theta =1$

4. $x^2{\cos }^2\theta –y^2{\sin }^2\theta =1$

Q. The ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and hyperbola $\frac{x^2}{A^2}-\frac{y^2}{B^2}=1$ are having a same foci and length of minor axis of ellipse is same as the conjugate axis of the hyperbola. If $e_1$ & $e_2$ are the eccentricities of ellipse and hyperbola respectively, then the value of $\frac{1}{e_1^2}+\frac{1}{e_2^2}$ is

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

Q. If the distance between the foci and the distance between two directrices of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are in the ratio $3:2$, then $b:a$ is

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

Q. Let $L$ is distance between two parallel normals of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b,$ then maximum value of $L$ is:-

1. $2a$
2. $2b$
3. $2(a-b)$
4. $2(a+b)$

Q. The foci of the hyperbola are $S(5,6),S^{\prime }(-3,-2)$. If its eccentricity is $2$, then the equation of its directrix corresponding to focus $S$ is

1. $x+y-3=0$
2. $x+y-5=0$
3. $x+y-1=0$
4. $x+y-7=0$

Q.

The combined equation of the asymptotes of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=-1$..

1. True

2. False

Q. The locus of the equation $x^2+y^2+z^2+1=0$ is

1. A sphere

2. An empty set

3. A degenerate set

4. A pair of planes

Q.

Show that the line $\frac{x}{a}+\frac{y}{b}=1$ touches the curve y = $b.e^{-\frac{x}{a}}$ at the point where the curve intersects the axis of Y.

Q. The length of the latus rectum of a parabola whose axis is parallel to the $y-$ axis and is passing through the points $(0,1),(1,2)$ and $(2,\frac{7}{2}),$ is

Q. How to find area of curve using definite integral?

Q. If $a,b,c$ are in $A.P$, the straight line $ax+by+c=0$, passes through a fixed point which lie on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{4}=1,$ then eccentricity of hyperbola is

Q.

If a chord joining $P(aSec\theta ,atan\theta ),Q(aSec\alpha ,atan\alpha )$ on the hyperbola $x^2-y^2=a^2$ is the normal at P, then $Tan\alpha$

1. 

2. 

3. 

4. 

Q. A hyperbola having the transverse axis of length $2\sin \theta$ unit, is confocal with the ellipse $3x^2+4y^2=12$, then its equation is

1. $x^2{\text{cosec}}^2\theta -y^2{\sec }^2\theta =1$
2. $x^2{\sec }^2\theta -y^2{\text{cosec}}^2\theta =1$
3. $x^2{\cos }^2\theta -y^2{\sin }^2\theta =1$
4. $x^2{\sin }^2\theta -y^2{\cos }^2\theta =1$

Q. The locus of the point of intersection of the lines $bxt-ayt=ab$ and $bx+ay=abt$, $t$ being parameter is

1. an ellipse with equation $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$
2. a hyperbola with equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
3. a hyperbola with equation $\frac{x^2}{b^2}-\frac{y^2}{a^2}=1$
4. an ellipse with equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Q. If the normal at $‘{\theta }^{\prime }$ on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ meets the transverse axis at G, then AG.A’G =

1. 
2. 
   
3. 
   
   
4. None

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

1. $\sqrt{3}$
2. Option c equation is not clear
3. Option a equation is not clear
4. Option b equation is not clear

Q. Let $e_1$ and $e_2$ be the eccentricities of the ellipse, $\frac{x^2}{25}+\frac{y^2}{b^2}=1(b<5)$ and the hyperbola, $\frac{x^2}{16}-\frac{y^2}{b^2}=1$ respectively satisfying $e_1{e}_2=1$. If $\alpha$ and $\beta$ are the distance between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair $(\alpha ,\beta )$ is equal to:

1. $(8,12)$
2. $(\frac{24}{5},10)$
3. $(\frac{20}{3},12)$
4. $(8,10)$

Q. For some $\theta \in (0,\frac{\pi }{2})$, if the eccentricity of the hyperbola, $x^2-y^2{\sec }^2\theta =10$ is $\sqrt{5}$ times the eccentricity of the ellipse, $x^2{\sec }^2\theta +y^2=5$, then the length of the latus rectum of the ellipse, is:

1. $\frac{4\sqrt{5}}{3}$
2. $\frac{2\sqrt{5}}{3}$
3. $2\sqrt{6}$
4. $\sqrt{30}$

Q. A hyperbola having the transverse axis of length $2\sin \theta$ unit, is confocal with the ellipse $3x^2+4y^2=12$, then its equation is

1. $x^2{\text{cosec}}^2\theta -y^2{\sec }^2\theta =1$
2. $x^2{\sec }^2\theta -y^2{\text{cosec}}^2\theta =1$
3. $x^2{\sin }^2\theta -y^2{\cos }^2\theta =1$
4. $x^2{\cos }^2\theta -y^2{\sin }^2\theta =1$

Q.

How do you write the standard form of a parabola with the vertex and focus?

Q.

In a hyperbola e = $\frac{9}{4}$ and the distance between the directrices is 3. Then the length of Transverse axis is

1. $\frac{27}{2}$

2. $\frac{27}{8}$

3. $\frac{27}{4}$

4. $\frac{17}{4}$

Q. $Insertfourgeometricmeansbetween160and54$

Q.

In a hyperbola the distance between the foci is 2 and the distance between directrices is 1 Its eccentricity is

1. 

2. 3/2

3. 

4. 5/2

Q. If $x+y=k$ is a normal to the parabola $y^2=12x$, $p$ is the length of the perpendicular from the focus of the parabola on this normal; then $\frac{3k^3+2p^2}{247}$ is equal to

Q. A double ordinate $PQ$ of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is such that $\Delta OPQ$ is equilateral, $O$ being the centre of the hyperbola. Then the eccentricity $e$ satisfies the relation

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

Q.

How do you find the exact value of the six trigonometric functions of $\theta$ when your given a point $(-4,-6)$?