Corresponding Points : Ellipse

Q.

The radius of a circle, having minimum area, which touches the curve$y=4-x^2$and the lines, $y=\left|x\right|$ is

1. $2(\sqrt{2}–1)$

2. $4(\sqrt{2}–1)$

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

4. $2(\sqrt{2}+1)$

Q.

Tangent and normal are drawn at $P\left(16,16\right)$ on the parabola $y^2=16x$, which intersect the axis of the parabola at $A$and $B$respectively. If $C$ is the center of the circle through the points $P,A,$and $B$and $\angle CPB=\theta$, then a value of $\tan \theta$is

1. $3$

2. $\frac{4}{3}$

3. $\frac{1}{2}$

4. $2$

Q. Let the point $P(\alpha ,\beta )$ on the ellipse $4x^2+3y^2=12,$ in the first quadrant such that the area enclosed by the lines $y=x,y=\beta ,x=\alpha$ and the $x-$ axis is maximum, then

1. $P\equiv (\frac{3}{2},1)$
2. $P\equiv (\frac{\sqrt{3}}{2},\sqrt{3})$
3. eccentric angle of $P$ is $\frac{\pi }{3}$
4. eccentric angle of $P$ is $\frac{\pi }{6}$

Q.

The point of intersection of the tangents at the point P on the ellipse$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and its corresponding point Q on the auxiliary circle, lies on the line

1. $x=\frac{a}{e}$

2. $x=0$

3. $y=0$

4. $Noneofthese$

Q. If $x,y\in R$, satisfies the equation $\frac{(x-4{)}^2}{4}+\frac{y^2}{9}=1,$ then the difference between the largest and smallest value of the expression $\frac{x^2}{4}+\frac{y^2}{9}$ is

Q. If the ratio of area of triangle inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ to that of triangle formed by the corresponding points on the auxiliary circle is $\frac{1}{2}$, then the eccentricity of the ellipse is

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

Q. If $x$ and $y$ are real numbers which satisfy the relation $x^2+9y^2-4x+6y+4=0,$ then the maximum value of $\frac{(4x-9y)}{2}$ is

Q. If $4x^2+y^2=1,$ then the maximum value of $12x^2-3y^2+16xy$ is