Equation of Circle in Complex Form

Q. The area of the region $\left\{\right(x,y):0\le x\le \frac{9}{4},0\le y\le 1,x\ge 3y,x+y\ge 2\}$ is

1. $\frac{11}{32}$ sq. units.
2. $\frac{35}{96}$sq. units.
3. $\frac{37}{96}$ sq. units.
4. $\frac{13}{32}$sq. units.

Q. Let $P$ and $Q$ be two distinct points on a circle which has centre at $C(2,3)$ and which passes through origin $O.$ If $OC$ is perpendicular to both the line segments $CP$ and $CQ$, then the set $\{P,Q\}$ is equal to

1. $\left\{\right(2+2\sqrt{2},3+\sqrt{5}),(2-2\sqrt{2},3-\sqrt{5}\left)\right\}$
2. $\left\{\right(4,0),(0,6\left)\right\}$
3. $\left\{\right(-1,5),(5,1\left)\right\}$
4. $\left\{\right(2+2\sqrt{2},3-\sqrt{5}),(2-2\sqrt{2},3+\sqrt{5}\left)\right\}$

Q. Let $z_1$ and $z_2$ be two complex numbers satisfying $\left|z_1\right|=9$ and $|z_2-3-4i|=4$. Then the minimum value of $|z_1-z_2|$ is :

1. $0$
2. $1$
3. $\sqrt{2}$
4. $2$

Q. The least value of $|z-3-4i{|}^2+|z+2-7i{|}^2+|z-5+2i{|}^2$ occurs when $z=x+iy$ then the value of $x$ is

Q.

The area of the circle centred at$(1,2)$and passing through $(4,6)$ is

1. $5\mathrm{\pi }\mathrm{sq}\mathrm{units}$

2. $10\mathrm{\pi }\mathrm{sq}\mathrm{units}$

3. $25\mathrm{\pi }\mathrm{sq}\mathrm{units}$

4. None of these

Q. Each of the circles $|z-1-i|=1$ and $|z-1+i|=1$ where $z=x+iy,$ touches internally a circle of radius $2$ units. The equation of the circle touching all the three circles can be

1. $3z\overline{z}+z+\overline{z}-1=0$
2. $3z\overline{z}-7z-7\overline{z}+15=0$
3. $z\overline{z}-z-\overline{z}-3=0$
4. $3z\overline{z}+z+\overline{z}+1=0$

Q. If $|z-1-i|=1$, then the locus of a point represented by the complex number $5(z-i)-6$ is

1. a circle with centre $(-1,0)$ and radius $5$
2. a line passing through the origin
3. a line passing through $(-1,0)$
4. a circle with centre $(1,0)$ and radius $3$

Q.

The equation $z\overline{z}+a\overline{z}+\overline{a}z+b$ = 0, b$\in$R represents a circle if

1. 

2. 

3. 

4. None of these

Q. The region of argand diagram defined by \( |z-1|+|z+1|\leq 4\) is

1. interior of an ellipse
2. exterior of a circle
3. interior and boundary of an ellipse
4. None of the above

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

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

Q. The largest value of $r$ for which the region represented by the set $\{\omega \in C:|\omega -4-i|\le r\}$ is contained in the region represented by the set $\{z\in C:|z-1|\le |z+i|\},$ is equal to:

1. $2\sqrt{2}$ unit
2. $\frac{3}{2}\sqrt{2}$ unit
3. $\sqrt{17}$ unit
4. $\frac{5}{2}\sqrt{2}$ unit

Q. The complex number having least positive argument and satisfying the inequality $|z-5i|\le 3$ is

1. $\frac{12}{5}+\frac{16}{5}i$
2. $\frac{12}{5}+\frac{12}{5}i$
3. $\frac{16}{5}+\frac{12}{5}i$
4. $\frac{3}{5}+\frac{4}{5}i$

Q.

The area of the circle whose centre is at $(1,2)$ and which passes through the point $(4,6)$is:

1. $5\pi$

2. $10\pi$

3. $25\pi$

4. None of these

Q. If $z=x+iy$ and $x^2+y^2=16$, then the range of $||x|-|y||$ is

1. $[0,4]$
2. $[0,2]$
3. None of these
4. $[2,4]$

Q.

In a triangle, $ABC,AD,BE$, and $CF$ are the altitudes, and $R$ is the circumradius then the radius of the circle $DEF$ is

1. $2R$

2. $R$

3. $\frac{R}{2}$

4. None of these

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 $Re\left(\frac{1}{z}\right)>\frac{1}{2}$ 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. 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. 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 $\text{Im}\left(\frac{2z+1}{iz+1}\right)=-3$, then locus of $z$ is

1. a circle
2. a parabola
3. a straight line
4. none of these

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

Q. If $|z|=1$ and $w=-1+3z$, then the locus of $w$ on argand plane is

1. a circle whose centre is at $(-1,0)$ and radius is $3$ units.
2. a circle whose centre is at $(-1,0)$ and radius is $9$ units.
3. a circle whose centre is at $(0,-1)$ and radius is $3$ units.
4. a circle whose centre is at $(1,0)$ and radius is $3$ units.

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 (where $a,b$ are complex numbers)

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

Q. For equation $|z-z_1{|}^2+|z-z_2{|}^2=a$ to represent a circle, which of the following is/are true?

1. Centre of the circle is $\frac{(z_1+z_2)}{4}.$
2. Centre of the circle is $\frac{(z_1+z_2)}{2}.$
3. The circle is real, if $2a\ge |z_1-z_2{|}^2.$
4. The circle is real, if $4a\ge |z_1-z_2{|}^2.$

Q. Let $A\left(z_1\right)$ be the point of intersection of curves $\arg (z-2+i)=\frac{3\pi }{4}$ and $\arg (z+\sqrt{3}i)=\frac{\pi }{3}.$ $B\left(z_2\right)$ is the point on $\arg (z+\sqrt{3}i)=\frac{\pi }{3}$ such that $|z_2-5|$ is minimum, and $C\left(z_3\right)$ is the centre of circle $|z-5|=3.$ If the area of triangle $ABC$ is $\sqrt{k}$ sq. units, then the value of $k$ is

Q. For equation $|z-z_1{|}^2+|z-z_2{|}^2=a$ to represents a circle, which of the following is/are true

1. Centre of the circle is $\frac{(z_1+z_2)}{4}.$
2. Centre of the circle is $\frac{(z_1+z_2)}{2}.$
3. The circle is real, if $2a\ge |z_1-z_2{|}^2.$
4. The circle is real, if $4a\ge |z_1-z_2{|}^2.$

Q.

Find the area of the region bounded by line x = 2 and parabola $y^2=8x$.

Q. If the complex number $z=x+iy$ satisfies the condition $|z+1|=1$ , then $z$ lies on

1. Circle with centre $(-1,0)$ and radius $1$
2. $y-$axis

3. none of these

4. $x-$axis