Auxiliary Circle of Ellipse

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. A line is such that its segments between the straight lines 5x – y = 4 and 3x + 4y – 4 = 0 is bisected at the points (1, 5). Its equation is

1. 83x – 35y + 92 = 0
2. None of these
3. 23x – 7y + 6 = 0
4. 7x + 4y + 3 = 0

Q. If the minimum area of the triangle formed by a tangent to the ellipse $\frac{x^2}{b^2}+\frac{y^2}{4a^2}=1$ and the coordinate axis is $kab$, then $k$ is equal to

Q. A circle touches the parabola $y^2=2x$ at $P(\frac{1}{2},1)$ and cuts the parabola at its vertex $V.$ If the centre of the circle is $Q,$ then

1. the radius of the circle is $\frac{5\sqrt{2}}{4}$
2. the radius of the circle is the maximum value of $\frac{1}{4}\sin 4x+\frac{7}{4}\cos 4x.$
3. area of $△PVQ$ is $\frac{15}{16}$
4. slope of $PQ$ is $-2$

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. $[2,3]$

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. The plane $\frac{x}{2}+\frac{y}{3}+\frac{z}{4}=1$ cuts the axes in $A,B,C$ then the area of the $△ABC$ (in $\text{sq.units}$) is

1. $\sqrt{59}$
2. $\sqrt{61}$
3. $\sqrt{63}$
4. $\sqrt{65}$

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 circle $x^2+y^2-6x-6y+9=0$ is inscribed in a triangle which has two of its sides along the coordinate axes. The locus of the circumcentre of the triangle is $x+y-\frac{b}{a+b}xy+a(x^2+y^2{)}^{\frac{1}{b}}=0$. Find $a+b$

Q. The area under the curve $y=x^2–3x+2$ with boundaries as x-axis and the ordinates x = 0, x = 3 is

1. $\frac{3}{8}$
2. $\frac{2}{3}$
3. $\frac{3}{5}$
4. $\frac{9}{2}$

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. 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.

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

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

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

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

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. The number of lattice points (lattice points are points with both coordinate integers) on the circle with centre (199, 0) and radius 199 is___

Q. If $A(2,8)$ is an interior point of a circle $x^2+y^2-2x+4y-P=0$ which neither touches nor intersects the axes, then set for $P$ is

1. $P<-1$
2. $P<-4$
3. $P>96$
4. $\varphi$

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 a line $\frac{x-x_1}{a_1}=\frac{y-y_1}{b_1}=\frac{z-z_1}{c_1}$ lies in a plane $a_{2}x+b_{2}y+c_{2}z=d,$ then which of the following is / are correct -

1. 
2. 
3. 
4. None of these

Q. The equation of tangent to the curve $y=1-e^{\frac{x}{2}}$ at the point where it meets y - axis is -

1. $x+2y=0$
2. $2x+y=0$
3. $x-y=2$
4. None of these

Q. If the tangent at P on the curve $x^m{y}^n=a^{m+n}$ meets the co-ordinates axes at A and B, then $AP:PB=$

1. $m^2:n^2$
2. $m^3:n^3$
   
3. $m:n$
   
4. $2m:n$
   

Q. Which of tangents to the curve $y=\cos (x+y),-2\pi \le x\le 2\pi$ is/are parallel to the line $x+2y=0$.

1. $2x+4y+3\pi =0$
2. $x+4y-\pi =0.$
3. $2x+4y-\pi =0.$
4. $x-4y-3\pi =0.$

Q. Show that the locus of the feet of the perpendiculars drawn from foci to any tangent of the ellipse is the auxiliary circle.

Q. If $\left(\sqrt{3}\right)bx+ay=2ab$ touches the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, then the eccentric angle of the point of contact is

1. $\frac{\pi }{4}$
2. $\frac{\pi }{3}$
3. $\frac{\pi }{6}$
4. $\frac{\pi }{2}$

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

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. 
2. 
3. 
4. 

Q. The equation of the tangent to the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ which is parallel to the line $y=3x$, is

1. $y=3x+\sqrt{241}$
2. $y=3x+13$
3. $y=3x+\sqrt{209}$
4. none of these

Q. Parallel to $x$-axis are $(a,\pm b)$.Find $a+b$

Q.

Which of the following is/are true?

1. Equation of auxiliary circle of ellipse,

2. For to be an ellipse a > b.
3. In the parametric representation of point where P lies on ellipse , is the angle made by P with respect to major axis.
4. Sum of focal distance of any point on the ellipse is a+b.

Q.

The line passing through the extremity A of the major axis and extremity B of the minor axis of the ellipse $x^2+9y^2=9$ meets its auxiliary circle at the point M. Then, the area (insqunits) of the triangle with vertices at A, M and the origin O is

1. $\frac{31}{10}$

2. $\frac{29}{10}$

3. $\frac{21}{10}$

4. $\frac{27}{10}$