Auxiliary Circle of Hyperbola

Q. The equation of rectangular hyperbola $xy=c^2$ is drawn by rotating $x^2-y^2=35$ and its asymptote is the coordinate axis, then value of $c$ is

1. $\pm \frac{35}{2}$
2. $\pm \sqrt{\frac{35}{2}}$
3. $\pm \sqrt{35}$
4. $\pm \frac{\sqrt{35}}{2}$

Q. $\mathrm{The}\mathrm{auxiliary}\mathrm{circle}\mathrm{of}\mathrm{the}\mathrm{hyperbola}\frac{x^2}{9}-\frac{y^2}{16}=1\mathrm{is}{x}^2+y^2=9$

1. False
2. True

Q.

A standard hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is drawn along with its auxiliary circle. A point P $(asec\theta ,btan\theta )$ is taken. A perpendicular is dropped from P to the x axis which meets at the axis at R. A tangent is drawn from R to auxiliary circle. Which angle is equal to θ

1. $\angle TOR$

2. $\angle TRO$

3. $\angle PRT$

4. $\angle TO{V}_1$

Q. The locus of foot of perpendicular from any focus of a hyperbola upon any tangent to the hyperbola is the auxiliary circle of the hyperbola. Consider the foci of a hyperbola as $(-3,-2)$ and $(5,6)$ and the foot of perpendicular from the focus $(5,6)$ upon tangent to the hyperbola as $(2,5).$ Then

1. radius of the auxiliary circle is $\sqrt{10}$
2. conjugate axis of the hyperbola is $2\sqrt{22}$
3. the directrix of the hyperbola corresponding to the focus $(5,6)$ is $2x+2y=11$
4. the point of contact of the tangent with the hyperbola is $(\frac{2}{3},9)$