Equation of Circle in Complex Form

Q. If $|Z-i|\le 2$ and $Z_1=5+3i$, then the maximum value of $|iZ+Z_1|$ is

1. $7+\sqrt{13}$
2. $7+\sqrt{12}$
3. $7$
4. $9$

Q.

The curve represented by the equation Re($\frac{1}{z}$) = 2 is:

1. Circle with centre (1/4, 0)

2. Straight line through (1/2, 0)

3. Ellipse

4. None of these

Q. If $z$ is a complex number lying in the fourth quadrant of Argand plane and $|\left[\frac{kz}{k+1}\right]+2i|>\sqrt{2}$ for all real values of $k(k\ne -1)$, then range of $\arg \left(z\right)$ is

1. $(-\frac{\pi }{8},0)$
2. $(-\frac{\pi }{6},0)$
3. $(-\frac{\pi }{4},0)$
4. None of these

Q. If $t$ and $c$ are two complex numbers such that $|t|\ne |c|,|t|=1$ and $z=\frac{at+b}{t-c},z=x+iy$, then locus of $z$ is a/an (where $a,b$ are complex numbers)

1. line segment
2. straight line
3. circle
4. ellipse

Q.

The centre of the circle $z\overline{z}-(2+3i)z-(2-3i)\overline{z}+9$ = 0 is

1. ( 2, -3 )

2. ( 2, 3 )

3. ( -2, -3 )

4. ( -2, 3 )

Q. Let $z$ be a complex number satisfying $|2z+10+10i|\le 5\sqrt{3}-5.$ If $\arg$ denotes the principal argument lying in $(-\pi ,\pi ],$ then the least value of $\arg \left(z\right)$ is

1. $\frac{-5\pi }{6}$
2. $\frac{3\pi }{4}$
3. $\frac{-3\pi }{4}$
4. $\frac{5\pi }{6}$

Q. Locus of complex number satisfying $\arg \left[\frac{(z-5+4i)}{(z+3-2i)}\right]=-\frac{\pi }{4}$ is the arc of a circle

1. whose radius is $5\sqrt{2}$ units
2. whose radius is $5$ units
3. whose length (of arc) is $\frac{15\pi }{\sqrt{2}}$ units
4. whose centre is $-2-5i$

Q. For positive constant $r,$ let $M$ be the set of complex numbers $z$ which satisfy $|z-4-3i|=r.$ Then which of the following statements is (are) CORRECT?

1. If $r=3,$ then the minimum value of $|z|$ for complex number $z$ which belongs to $M$ is $2.$
2. If $r=3,$ then the maximum value of $|z|$ for complex number $z$ which belongs to $M$ is $8.$
3. If $r=5,$ then the complex number having least modulus which belongs to $M$ is $z=0$
4. If $r=5,$ then the complex number having greatest modulus which belongs to $M$ is $z=8+6i$

Q. Let $A\left(z_1\right)$ and $B\left(z_2\right)$ be two points lying on the curve $z-3-4i=\frac{25}{\overline{z}-3+4i}$ where $|z_1|$ is maximum. Now, $A\left(z_1\right)$ is rotated about the origin in anti-clockwise direction through ${90}^{\circ }$ reaching at $P\left(z_0\right).$ If $A,B$ and $P$ are collinear, then the value of $|z_0-z_1||z_0-z_2|$ is

Q. Let $C_1$ and $C_2$ are concentric circles of radius $1$ unit and $8/3$ unit , respectively, having centre at $(3,0)$ on the Argand plane. If the complex number $z$ satisfies the inequality
${\log }_{1/3}\left(\frac{|z-3{|}^2+2}{11|z-3|-2}\right)>1$ then

1. $z$ lies outside $C_1$ but inside $C_2$
2. $z$ lies inside $C_1$
3. $z$ lies outside $C_2$
4. none of these

Q.

If the sets $A$ and $B$ are defined as

$A=\left\{\right(x,y):y=\frac{1}{x},x\in R-\{0\left\}\right\}$

$B=\left\{\right(x,y):y=-x,x\in R\}$, then

1. $A\cap B=A$

2. $A\cap B=B$

3. $A\cap B=\varnothing$

4. None of these

Q. If $\left|\frac{z-z_1}{z-z_2}\right|=3$, where $z_1$ and $z_2$ are fixed complex numbers and $z$ is a variable complex number, then '$z$' lies on a

1. Circle with centre as $\frac{9z_1-z_2}{8}$
2. Circle with ${}^{\prime }{z}_2^{\prime }$ as its interior point
3. Circle with centre as $\frac{9z_2-z_1}{8}$
4. Circle with ${}^{\prime }{z}_2^{\prime }$ as its exterior point

Q. Diameter of the circle given by $|(z-\alpha )/(z-\beta )|=k,k\ne 1$ , where $\alpha ,\beta$ are fixed points and $z$ is varying point in argand plane is

1. $\frac{k|\alpha -\beta |}{|1-k^2|}$
2. $\frac{2k|\alpha -\beta |}{|1-k^2|}$
3. $\frac{3k|\alpha -\beta |}{|1-k^2|}$
4. $\frac{4k|\alpha -\beta |}{|1-k^2|}$

Q. If a triangle $ABC$ is formed by the points represented by $z,$ $iz$ and $z+iz$ then

1. $ABC$ is an equilateral triangle
2. $ABC$ is an isosceles right angled triangle
3. Area of $△ABC$ is $\frac{1}{2}|z{|}^2$ sq. units
4. Area of $△ABC$ is $\frac{\sqrt{3}}{4}|z{|}^2$ sq. units

Q. Given that the two curves $\arg \left(z\right)=\frac{\pi }{6}$ and $|z-2\sqrt{3}i|=r$ intersect in two distinct points, then
([r] represents integaral part of $r$)

1. $\left[r\right]=3$
2. $2<r<3$
3. $\left[r\right]=2$
4. $3<r<2\sqrt{3}$

Q. If $Re\left(\frac{1}{z}\right)>\frac{1}{2}$ and $Re\left(z\right)>0$, then which of the following is/are true about the locus of $z$?

1. The locus of $z$ is the region inside a circle.
2. The locus of $z$ is the region inside an ellipse.
3. Area of locus is $\pi \text{sq. units}$
4. Centre of the locus is $(1,0)$

Q. Equation of tangent drawn to circle $|z|=r$ at the point $A\left(z_0\right)$ is

1. $\text{Re}\left(\frac{z}{z_0}\right)=1$
2. $z{\overline{z}}_0-z_0\overline{z}=2r^2$
3. $\text{Im}\left(\frac{z}{z_0}\right)=1$
4. $z{\overline{z}}_0+z_0\overline{z}=2r^2$

Q. If $x+y+z=1$, then the least value of $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$, is

1. $9$
2. $1$
3. $3$
4. $27$

Q. $z_1$ and $z_2$ lies on the circle with centre at the origin. The point of intersection $z_3$ of the tangents at $z_1$ and $z_2$ is given by

1. $\frac{1}{2}({\overline{z}}_1+{\overline{z}}_2)$
2. $\frac{2z_1{z}_2({\overline{z}}_2-{\overline{z}}_1)}{z_1{\overline{z}}_2-z_2{\overline{z}}_1}$
3. $\frac{1}{2}(\frac{1}{z_1}+\frac{1}{z_2})$
4. $\frac{z_1+z_2}{{\overline{z}}_1{\overline{z}}_2}$

Q. If $z$ is a complex number satisfying $|z|=1$, then the range of $\arg \left(\frac{1}{1-z}\right)$ is

1. $(0,\pi )$
2. $[0,\pi ]$
3. $[0,\frac{\pi }{2}]$
4. $(-\frac{\pi }{2},\frac{\pi }{2})$

Q.

If $|$z$|$ = 3, then the points representing the complex numbers $-2+4z$ lie on a

1. Straight line

2. Circle

3. Ellipse

4. Parabola

Q. Let complex numbers $\alpha$ and $\frac{1}{\overline{\alpha }}$ lie on circles $(x-x_0{)}^2+(y-y_0{)}^2=r^2$ and $(x-x_0{)}^2+(y-y_0{)}^2=4r^2$, respectively. If $z_0=x_0+i{y}_0$ satisfies the equation $2|z_0{|}^2=r^2+2$, then $|\alpha |=$

1. $\frac{1}{\sqrt{2}}$
2. $\frac{1}{2}$
3. $\frac{1}{\sqrt{7}}$
4. $\frac{1}{3}$