Director Circle: Ellipse

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. Let an ellipse having major axis and minor axis parallel to the $x-$axis and the $y-$axis respectively. Its two foci $S$ and $S^{\prime }$ are $(3,4)$, $(5,4)$ and the line $x-3y+17=0$ is a tangent to the ellipse at a point $P.$ Then

1. Eccentricity of the ellipse is $\frac{1}{3}$
2. Length of semi-minor axis is $2\sqrt{2}$
3. The radius of the director circle of the ellipse is $4$
4. The equation of the one of the directrices is $x=9$

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

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

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. 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. Area of the greatest rectangle that can be inscribed in an ellipse $9x^2+16y^2=144$ is equal to $($in sq. units$)$

Q. Draw a rough sketch of the region $\left\{\right(x,y)$ : $y^2\le 6ax\&x^2+y^2\le 16a^2\}$. Also find the area of the region sketched using method of integration.

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. S(3, 4) and S’(9, 12) are the foci of an ellipse and the foot of the perpendicular from S to a tangent to the ellipse is (1, –4). Then the eccentricity of the ellipse is

1. 
2. 
3. 
4. none

Q. Find the equation to the locus of a point which moves so that the sum of its distance from $(3,0)$ and $(-3,0)$ is less than $9$.

Q. Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, $\frac{x^2}{4}+\frac{y^2}{2}=1$ from any of its foci?

1. $(-1,\sqrt{3})$
2. $(-2,\sqrt{3})$
3. $(-1,\sqrt{2})$
4. $(1,2)$

Q. The equation to the curve which is such that portion of the axis of $x$ cut off between the origin and the tangent at any point is proportional to the ordinate of that point is:

1. $x^2=y(C-K\log y)$
2. $x=y(C-K\log y)$
3. none of these
4. $\log x=K{y}^2+C$

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. If the squares of the lengths of the tangents from a point $P$ to the circles $x^2+y^2=a^2,x^2+y^2=b^2$ and $x^2+y^2=c^2$are in A.P., then

1. $a^2,b^2,c^2$ are in A.P
2. $\frac{1}{a^2},\frac{1}{b^2},\frac{1}{c^2}$ are in H.P
3. $a^2,b^2,c^2$ are in G.P
4. $\frac{1}{a^2},\frac{1}{b^2},\frac{1}{c^2}$ are in A.P

Q. Global minimum value of $y=f\left(x\right)$ for $x\in [0,2]$ is

1. $7$
2. $5$
3. $8$
4. $9$

Q.

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

Q. If the length of the tangent drawn from any point on the circle $x^2+y^2+15x-17y+c^2=0$ to the circle $x^2+y^2+15x-17y+21=0is\sqrt{5}$ units , then $c$ is equal to

1. $-3$
2. $3$
3. $4$
4. $-4$

Q. $PQRS$ is a square. $SR$ is a tangent (at point $S$) to the circle with centre $O$ and $TR=OS.$ Then find the ratio of area of the circle to the area of the square.

1. $\frac{\pi }{\sqrt{3}}$
2. $\frac{\pi }{3}$
3. $\frac{\pi }{\sqrt{2}}$
4. $\frac{\sqrt{3}}{\pi }$

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

Q. Using integration find the area of the triangle formed by positive x-axis and tangent and normal to the circle $x^2+y^2=4$ at $(1,\sqrt{3})$.