Director Circle: Ellipse

Q.

Locus of the point of intersection of the tangents which meet at right angles is called.

1. Auxiliary circle

2. None of these

3. Directrix

4. Director circle

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. If common tangents of $x^2+y^2=r^2$ and $\frac{x^2}{16}+\frac{y^2}{9}=1$ form a square, then the length of the diagonal of the square is

1. $10$
2. $25$
3. $5r$
4. $5\sqrt{2}r$

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.

Locus of the point of intersection of the tangents which meet at right angles is called.

1. Directrix

2. Director circle

3. Auxiliary circle

4. None of these

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. $x^2+y^2=81$