Auxiliary Circle of Hyperbola

Q. If ‘CF’ is the perpendicular from the centre C of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ on the tangent at any point P and G is the point where the normal at P meets the major axis, then CF.PG is

1. 
2. 
3. 
4. 

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. $\angle TO{V}_1$