Director Circle: Ellipse

Q.

The area of the smaller segment cut off from the circle ${\mathrm{x}}^2+{\mathrm{y}}^2=9$ by $\mathrm{x}=1$ is

1. $\left[\left(\frac{1}{2}\right)\left(9{\sec }^{-1}\left(3\right)-\sqrt{8}\right)\right]\mathrm{sq}\mathrm{units}$

2. $\left(9{\sec }^{-1}\left(3\right)-\sqrt{8}\right)\mathrm{sq}\mathrm{units}$

3. $\left(\sqrt{8}-9{\sec }^{-1}\left(3\right)\right)\mathrm{sq}\mathrm{units}$

4. None of these

Q. The area of the director circle of the ellipse $\frac{x^2}{5}+\frac{y^2}{4}=1$ is

1. $5\pi \text{sq. units}$
2. $4\pi \text{sq. units}$
3. $9\pi \text{sq. units}$
4. $18\pi \text{sq. units}$

Q. For all real $p\in [-1,1],$ the line $2px+y\sqrt{1-p^2}=1$ touches a fixed ellipse whose axes are coordinate axes.
Then which of the following is/are true

1. Foci of the ellipse are $(0,\pm \frac{1}{2})$
2. Foci of the ellipse are $(0,\pm \frac{\sqrt{3}}{2})$
3. Director circle of the ellipse is $x^2+y^2=\frac{3}{4}$
4. Director circle of the ellipse is $x^2+y^2=\frac{5}{4}$

Q. Equation of director circle of the ellipse $\frac{x^2}{3}+\frac{y^2}{6}=1$ is

1. $x^2+y^2=36$
2. $x^2+y^2=9$
3. $x^2+y^2=45$
4. $x^2+y^2=81$