Definition of Hyperbola

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

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. 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 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.

An equation $\frac{x^2}{p}+\frac{y^2}{q}=1$ represents a hyperbola is p and q are of the opposite signs.

1. True

2. False