Auxiliary Circle of Ellipse

Q.

The locus of the point of intersection of the lines $axsec\theta +bytan\theta =aandaxtan\theta +bysec\theta =b$, where $\theta$ is the parameter, is

1. A straight line

2. A circle

3. An ellipse

4. A hyperbola

Q. The tangent at any point on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the auxiliary circle at two points which subtend a right angle at the centre. When the eccentricity of the ellipse is minimum then

1. the $y-$ intercept made by the tangent is $y=\pm b\sqrt{2}$ when $a>b$
2. the $x-$ intercept made by the tangent is $x=\pm a\sqrt{2}$ when $a<b$
3. minimum value of eccentricity is $\frac{1}{\sqrt{2}}$
4. When $a>b$, then the area enclosed between the tangents and the ellipse is $(4-\pi )ab$

Q. The auxiliary circle of family of ellipses, passes through origin and makes intercept of $8$ and $6$ units on the $x-$ axis and the $y-$ axis respectively. If eccentricity of all such family of ellipse is $\frac{1}{2}$, then the locus of focus of ellipse will be

1. $4x^2+4y^2+32x-24y+75=0$
2. $4x^2+4y^2-32x-24y+75=0$
3. $4x^2+4y^2-32x+24y+75=0$
4. $4x^2+4y^2-32x-24y-75=0$

Q.

Equation of the locus of the pole with respect to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ of any tangent line to the auxiliary circle is the curve
$\frac{x^2}{a^4}+\frac{y^2}{b^4}={\lambda }^2$ where

1. ${\lambda }^2=a^2$

2. ${\lambda }^2=\frac{1}{a^2}$

3. ${\lambda }^2=b^2$

4. ${\lambda }^2=\frac{1}{b^2}$

Q. If the ellipse $\frac{x^2}{4}+y^2=1$ meets the ellipse $x^2+\frac{y^2}{a^2}=1$ in four distinct points and $a=b^2-5b+7,$ then $b$ does not lie in

1. $[4,5]$
2. $(-\infty ,2)\cup (3,\infty )$
3. $(-\infty ,0)$
4. $[2,3]$