Director Circle: Hyperbola

Q.

For a standard hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Match the following.

$\begin{array}{cc}\text{Column 1}& \text{Column 2}\\ 1.a^2>b^2& \text{P.Director circle is real}\\ 2.a^2=b^2& \text{Q.Director circle is imaginary}\\ 3.a^2<b^2& \text{R.Centre is the only point from which}\\ & \text{two perpendicular tangents can be drawn on the}\\ & \text{hyperbola}\end{array}$

1. $1-P,2-Q,3-R$

2. $1-R,2-Q,3-P$

3. $1-P,2-R,3-Q$

4. $1-Q,2-P,3-R$

Q. Which among the following option(s) is/are correct for hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, where $n$ is the number of points on the plane through which perpendicular tangents are drawn to it and $e$ is the eccentricity.

1. If $n=1$, then $e=\sqrt{2}$
2. If $n>1$, then $0<e<\sqrt{2}$
3. If $n=0$, then $e>\sqrt{2}$
4. If $n=0$, then $e>2$

Q.

Find the equation to the director circle of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

1. $x^2+y^2=a^2-b^2$

2. $x^2-y^2=a^2-b^2$

3. $x^2+y^2=a^2+b^2$

4. $x^2-y^2=a^2+b^2$

Q.

Find the equation to the director circle of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

1. $x^2+y^2=a^2-b^2$

2. $x^2-y^2=a^2-b^2$

3. $x^2+y^2=a^2+b^2$

4. $x^2-y^2=a^2+b^2$

Q. The perpendicular tangents drawn to the hyperbola
$\frac{x^2}{25}-\frac{y^2}{16}=1$
intersect on the curve $x^2+y^2=9$

1. False
2. True

Q. The perpendicular tangents drawn to the hyperbola
$\frac{x^2}{25}-\frac{y^2}{16}=1$
intersect on the curve $x^2+y^2=9$

1. False
2. True