Auxiliary Circle of Hyperbola

Q. Equation of the circle with $AB$ as its diameter is

1. $x^2+y^2-24x-12=0$
2. $x^2+y^2+24x-12=0$
3. $x^2+y^2–12x+24=0$
4. $x^2+y^2+12x+24=0$

Q. The locus of foot of perpendicular from any focus of a hyperbola upon any tangent to the hyperbola is the auxiliary circle of the hyperbola. Consider the foci of a hyperbola as $(-3,-2)$ and $(5,6)$ and the foot of perpendicular from the focus $(5,6)$ upon tangent to the hyperbola as $(2,5).$ Then

1. radius of the auxiliary circle is $\sqrt{10}$
2. conjugate axis of the hyperbola is $2\sqrt{22}$
3. the directrix of the hyperbola corresponding to the focus $(5,6)$ is $2x+2y=11$
4. the point of contact of the tangent with the hyperbola is $(\frac{2}{3},9)$

Q. The equation of auxiliary circle of hyperbola $25y^2+250y-16x^2-32x+209=0$ is

1. $(x+1{)}^2+(y+5{)}^2=25$
2. $x^2+y^2=25$
3. $x^2+y^2=16$
4. $(x+1{)}^2+(y+5{)}^2=16$

Q.

For all four circles $M,N,O$ and $P$, the following four equations are given:

Circle $M:x^2+y^2=1$

Circle $N:x^2+y^2-2x=0$

Circle $O:x^2+y^2-2x-2y+1=0$

Circle $P:x^2+y^2-2y=0$

If the centre of circle $M$ is joined with the centre of the circle $N$, further centre of circle $N$ is joined with the centre of the circle $O$, centre of circle $O$ is joined with the centre of circle $P$ and lastly, the centre of circle $P$ is joined with the centre of circle $M$, then these lines form the sides of $a$:

1. Rectangle

2. Square

3. Parallelogram

4. Rhombus

Q. The equation of the auxiliary circle of the hyperbola $\frac{x^2}{64}-\frac{y^2}{36}=1$ is

1. $x^2+y^2=100$
2. $x^2+y^2=28$
3. $x^2+y^2=64$
4. $x^2+y^2=36$

Q.

The radius of the circle $x^2+y^2+4x+6y+13=0$is

1. $\sqrt{26}$

2. $\sqrt{13}$

3. $\sqrt{23}$

4. $0$

Q. Equation of hyperbola whose axes are parallel to coordinate axis with $e=\sqrt{2}$ and having distance between the foci as $1$ unit is :

1. $y^2-x^2=\frac{1}{8}$
2. $x^2-y^2=5$
3. $y^2-x^2=5$
4. $x^2-y^2=\frac{1}{8}$

Q. Taking axes of hyperbola as coordinate axes, find its equation when the distance between the foci is $16$ and eccentricity is $\sqrt{2}$

1. $x^2-y^2=16$
2. $x^2-y^2=32$
3. $x^2-y^2=8$
4. $x^2-y^2=64$

Q. Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in following cases :
The distance between the foci $=16$ and eccentricity $=\sqrt{2}$

Q.

How many asymptotes does a hyperbola have?

Q. The equation of auxiliary circle of hyperbola $25y^2+250y-16x^2-32x+209=0$ is

1. $(x+1{)}^2+(y+5{)}^2=25$
2. $x^2+y^2=25$
3. $x^2+y^2=16$
4. $(x+1{)}^2+(y+5{)}^2=16$

Q. Let $E$ be an standard ellipse (center at the origin and major axis as $x-$axis) whose length of major axis is $2a$ and length of minor axis is $2b$. Let $C$ be a circle with centre at origin and cutting the ellipse at an angle $\alpha$, then the value of radius of $C$ for which the angle $\alpha$ is maximum is

1. $\sqrt{\frac{a^2-b^2}{2}}$
2. $\sqrt{\frac{a^2+b^2}{2}}$
3. $\sqrt{a^2-b^2}$
4. $\frac{1}{2}\sqrt{a^2-b^2}$

Q.

A standard hyperbola is given as below. Which among the points given would lie on the auxiliary circle of the hyperbola?

1. 

2. A

3. B

4. O

Q. If $P$ and $Q$ are two points on the hyperbola $\frac{x^2}{5}-\frac{y^2}{8}=1$ whose centre is $C$ such that $CP$ is perpendicular to $CQ,$ then the value of $\frac{1}{(CP{)}^2}+\frac{1}{(CQ{)}^2}$ is equal to

1. $\frac{3}{20}$
2. $\frac{5}{40}$
3. $\frac{13}{40}$
4. $\frac{3}{40}$

Q.

What is the equation of the auxiliary circle of a hyperbola
$\frac{x^2}{16}-\frac{y^2}{9}=1$

1. $x^2+y^2=9$

2. $x^2+y^2=16$

3. $x^2+y^2=25$

4. $x^2+y^2=1$

Q. Three equal circle of unit radius touch each other. then the area of the circle circumscribing the three circle is:

1. $6\pi (2+\sqrt{3}{)}^2$
2. $\frac{\pi }{6}(2+\sqrt{3}{)}^2$
3. $\frac{\pi }{3}(2+\sqrt{3}{)}^2$
4. $3\pi 2+\sqrt{3}{)}^2$

Q.

A standard hyperbola is given as below. Which among the points given would lie on the auxiliary circle of the hyperbola?

1. A

2. B

3. O

4. $S_1$

Q. The equation of the auxiliary circle of the hyperbola $\frac{x^2}{64}-\frac{y^2}{36}=1$ is

1. $x^2+y^2=100$
2. $x^2+y^2=28$
3. $x^2+y^2=64$
4. $x^2+y^2=36$

Q.

If a circle cuts a rectangular hyperbola $xy=c^2$ in A, B, C, D and the parameters of these four points be $t_1,t_2,t_3$ and $t_4$ respectively. Then if the eccentricity of a conic is equal to $t_1.t_2.t_3.t_4$, the conic can't be

1. Circle
2. Parabola
3. Ellipse
4. Hyperbola

Q.

A standard hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is drawn along with its auxiliary circle. A point P $(asec\theta ,btan\theta )$ is taken. A perpendicular is dropped from P to the x axis which meets at the axis at R. A tangent is drawn from R to auxiliary circle. Which angle is equal to θ

1. 

2. 

3. 

4. 

Q. A is the vertex and S is the focus of the parabola $y^2=4ax$. Prove that the common chord of the parabola and the circle, centre A and radius equal to 3/8th of the length of the latus rectum of the parabola is the perpendicular bisector of AS.

Q. The equation of circle described on the straight line joining the points $(4,\frac{\pi }{6})$ and $(3,\frac{\pi }{3})$ as diameter is

1. $r^2+r[4\cos (\theta -\frac{\pi }{6})+3\cos (\theta -\frac{\pi }{3})]+6\sqrt{3}=0$
2. $r^2-r[4\cos (\theta -\frac{\pi }{6})+3\cos (\theta -\frac{\pi }{3})]+6\sqrt{3}=0$
3. $r^2+r[4\cos (\theta -\frac{\pi }{6})-3\cos (\theta -\frac{\pi }{3})]+6\sqrt{3}=0$
4. $r^2-r[4\cos (\theta -\frac{\pi }{6})+3\cos (\theta -\frac{\pi }{3})]-6\sqrt{3}=0$

Q. $\mathrm{The}\mathrm{auxiliary}\mathrm{circle}\mathrm{of}\mathrm{the}\mathrm{hyperbola}\frac{x^2}{9}-\frac{y^2}{16}=1\mathrm{is}{x}^2+y^2=9$

1. False
2. True

Q. Equation of hyperbola whose axes are parallel to coordinate axis with $e=\sqrt{2}$ and having distance between the foci as $1$ unit is :

1. $y^2-x^2=\frac{1}{8}$
2. $x^2-y^2=5$
3. $y^2-x^2=5$
4. $x^2-y^2=\frac{1}{8}$

Q.

What is the equation of the auxiliary circle of a hyperbola
$\frac{x^2}{16}-\frac{y^2}{9}=1$

1. $x^2+y^2=9$

2. $x^2+y^2=16$

3. $x^2+y^2=25$

4. $x^2+y^2=1$

Q. The equation of the auxiliary circle of the hyperbola $\frac{x^2}{64}-\frac{y^2}{36}=1$ is

1. $x^2+y^2=100$
2. $x^2+y^2=28$
3. $x^2+y^2=64$
4. $x^2+y^2=36$

Q. If $y=2x-3$ is a tangent to the parabola $y^2=4a(x-\frac{1}{3})$, then $a$ is equal to:

1. $-1$
2. $0$
3. $\frac{-14}{3}$
4. $1$

Q. Equation of hyperbola whose axes are parallel to coordinate axis with $e=\sqrt{2}$ and having distance between the foci as $1$ unit is :

1. $y^2-x^2=\frac{1}{8}$
2. $x^2-y^2=5$
3. $y^2-x^2=5$
4. $x^2-y^2=\frac{1}{8}$

Q. If from any point on the circle $x^2+y^2+2gx+2fy+c=0$ tangents are drawn to the circle $x^2+y^2+2gx+2fy+c{\sin }^2\alpha +(g^2+f^2){\cos }^2\alpha =0$, then the angle between the tangents is

1. $2\alpha$
2. $3\alpha$
3. None of these
4. $\alpha$

Q.

What is the equation of the auxiliary circle of a hyperbola
$\frac{x^2}{16}-\frac{y^2}{9}=1$

1. $x^2+y^2=1$

2. $x^2+y^2=16$

3. $x^2+y^2=25$

4. $x^2+y^2=9$