Corresponding Points : Ellipse

Q.

A circle is inscribed in an equilateral triangle of side a, the area of any square inscribed in the circle is

1. 

2. 

3. 

4. 

Q. Let $A,B,C$ and $D$ be four points on the ellipse
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b,$ such that $AB$ and $CD$ cut its major axis at two distinct equidistant points from centre.
Let $\alpha ,\beta ,\gamma ,\delta$ represent eccentric angles of $A,B,C$ and $D$ respectively. Then
$\tan \frac{\alpha }{2}\tan \frac{\beta }{2}\tan \frac{\gamma }{2}\tan \frac{\delta }{2}=$

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. If $S$ and $S^{\prime }$ are the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1,$ and $P$ is any point on it then range of values of $SP\cdot S^{\prime }P$ is

1. $[9,16]$
2. $[9,25]$
3. $[16,25]$
4. $[1,10]$

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

1. None of the above
2. $(\sqrt{3},0)$
3. $(\frac{3}{2},1)$
4. $(\frac{1}{2},\sqrt{\frac{11}{3}})$