Auxiliary Circle of Ellipse

Q.

Function $f\left(x\right)=\left|x-\left(\frac{1}{2}\right)\right|+|x-1|+\tan x$ how many points are not differentiable in $(0,2)$?

1. $1$

2. $2$

3. $3$

4. $4$

Q. Parametric coordinates of a point on ellipse, whose foci are $(-1,0)$ and $(7,0)$ and eccentricity is $\frac{1}{2},$ is

1. $(8+3\cos \theta ,4\sqrt{3}\sin \theta )$
2. $(8-3\cos \theta ,4\sqrt{3}\sin \theta )$
3. $(3+8\cos \theta ,-2\sqrt{3}\sin \theta )$
4. $(3+8\cos \theta ,4\sqrt{3}\sin \theta )$

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

Q. If $\frac{x^2}{2}+\frac{y^2}{1}=1$ is an ellipse with foci $S_1$ and $S_2$ .Rectangle $S_{1}P{S}_{2}Q$ is completed where $P$ and $Q$ are points on ellipse then which among the following options are correct

1. Number of such pair $P,Q$ is four
2. Area of rectangle $S_{1}P{S}_{2}Q$ is $2$ sq. unit
3. There will be infinite such pairs $P,Q$
4. Rectangle $S_{1}P{S}_{2}Q$ is a square

Q. The auxiliary circle of $\frac{x^2}{9}+\frac{y^2}{4}=1$ touches a parabola having axis of symmetry as $y-$axis at its vertex . If the latus rectum of parabola is equal to radius of director circle of auxiliary circle, then the equation of parabola can be

1. $x^2=\sqrt{18}(y-3)$
2. $x^2=\sqrt{16}(y+3)$
3. $x^2=-\sqrt{16}(y-3)$
4. $x^2=-\sqrt{18}(y+3)$

Q. If $\theta +\pi$ is the eccentric angle of a point on the ellipse $16x^2+25y^2=400,$ then the corresponding point on the auxilary circle is

1. $(-4\cos \theta ,-4\sin \theta )$
2. $(-5\cos \theta ,-5\sin \theta )$
3. $(4\cos \theta ,4\sin \theta )$
4. $(5\cos \theta ,5\sin \theta )$

Q. If the equation $|z-a|+|z-b|=3$ represents an ellipse and $a,b\in R$, (where $a$ is fixed point), then $b$ lies

1. Outside the circle $|z-a|=3$
2. Inside the circle $|z-a|=3$
3. On the circumference of the circle $|z-a|=3$
4. None of these

Q.

Let $Q(acos\theta ,bsin\theta )$ be a point on the auxiliary circle. Then the corresponding point with respect to Q on the ellipse when a line drawn perpendicular to major axis AA' will be.

1. $(acos\theta ,asin\theta )$
2. $(acos\theta ,bsin\theta )$
3. $(bcos\theta ,asin\theta )$
4. $(bcos\theta ,bsin\theta )$

Q. The locus of the image of the focus of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ with respect to any of the tangents to the ellipse is

1. $(x+4{)}^2+y^2=100$
2. $(x+2{)}^2+y^2=50$
3. $(x-4{)}^2+y^2=100$
4. $(x-2{)}^2+y^2=50$

Q. If $\theta +\pi$ is the eccentric angle of a point on the ellipse $16x^2+25y^2=400,$ then the corresponding point on the auxilary circle is

1. $(-4\cos \theta ,-4\sin \theta )$
2. $(-5\cos \theta ,-5\sin \theta )$
3. $(4\cos \theta ,4\sin \theta )$
4. $(5\cos \theta ,5\sin \theta )$

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

Q.

Let $Q(acos\theta ,bsin\theta )$ be a point on the auxiliary circle. Then the corresponding point with respect to Q on the ellipse when a line drawn perpendicular to major axis AA' will be.

1. $(acos\theta ,asin\theta )$
2. $(acos\theta ,bsin\theta )$
3. $(bcos\theta ,asin\theta )$
4. $(bcos\theta ,bsin\theta )$

Q. The auxiliary circle of $\frac{x^2}{9}+\frac{y^2}{4}=1$ touches a parabola having axis of symmetry as $y-$axis at its vertex . If the latus rectum of parabola is equal to radius of director circle of auxiliary circle, then the equation of parabola can be

1. $x^2=\sqrt{18}(y-3)$
2. $x^2=\sqrt{16}(y+3)$
3. $x^2=-\sqrt{16}(y-3)$
4. $x^2=-\sqrt{18}(y+3)$

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