Conjugate Hyperbola

Q. The equation of a hyperbola, conjugate to the hyperbola $x^2+3xy+2y^2+2x+3y+1=0$, is

1. $x^2+3xy+2y^2+2x+3y+1=0$
2. $x^2+3xy+2y^2+2x+3y+2=0$
3. $x^2+3xy+2y^2+2x+3y+3=0$
4. $x^2+3xy+2y^2+2x+3y+4=0$

Q. The equation of the conjugate hyperbola of the hyperbola $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y-3=0$ is

1. $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y+2=0$
2. $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y+5=0$
3. $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y+3=0$
4. $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y-2=0$

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

Q. If the angle between the asymptotes of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $\frac{\pi }{3}.$ Then the eccentricity of conjugate 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. $e\ge \frac{2}{\sqrt{3}}$
2. $1<e<2$
3. $e<4$
4. $1<e<\sqrt{2}$