Equation of Circle in Complex Form

Q.

The area (in sq units) of the triangle formed by the points $(2,2)$, $(5,5)$ and $(6,7)$is

1. $\frac{9}{2}$

2. $5$

3. $10$

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

5. $14$

Q.

If z lies on the circle $|z-1|=1$, then $\frac{z-2}{z}$ equals

1. 0

2. 2

3. -1

4. None of these

Q. If two sides of a triangle are the roots of $x^2-7x+8=0$ and the angle between these sides is $\frac{\pi }{3},$ then the product of inradius and circumradius of the triangle is

1. $\frac{8}{7}$
2. $\frac{5}{3}$
3. $\frac{5\sqrt{2}}{3}$
4. $8$

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.

$7^{2{\log }_{7}5}=$

1. ${\log }_{7}35$

2. $5$

3. $25$

4. ${\log }_{7}25$

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

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

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. If $|z|=1$ and $|\omega -1|=1$ where $z,\omega \in C,$ then the largest set of values of $|2z-1{|}^2+|2\omega -1{|}^2$ equals

1. $[1,9]$
2. $[2,6]$
3. $[2,12]$
4. $[2,18]$

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

Q. The equation $z\overline{z}+a\overline{z}+\overline{a}z+b=0,bϵR$ represents a circle if

1. $|a{|}^2=b$
2. $|a{|}^2>b$
3. $|a{|}^2<b$
4. None of these

Q. Intercept made by the circle $z\overline{z}+\overline{a}z+a\overline{z}+r=0$ on the real axis on complex plane is

1. $\sqrt{(a+\overline{a})-r}$
2. $\sqrt{(a+\overline{a}{)}^2-2r}$
3. $\sqrt{(a+\overline{a}{)}^2-4r}$
4. $\sqrt{(a+\overline{a}{)}^2-8r}$

Q. If a complex number $z$ satisfies $|2z+10+10i|\le 5\sqrt{3}-5,$ then the least principle argument of $z,$ is

1. $-\frac{5\pi }{6}$
2. $0$
3. $-\frac{3\pi }{4}$
4. $-\frac{2\pi }{3}$

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

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

Q. The differential equations satisfied by the system of parabolas $y^2=4a(x+a)$ is:

1. $y{\left(\frac{dy}{dx}\right)}^2-2x\left(\frac{dy}{dx}\right)-y=0$
2. $y{\left(\frac{dy}{dx}\right)}^2+2x\left(\frac{dy}{dx}\right)-y=0$
3. $y\left(\frac{dy}{dx}\right)+2x\left(\frac{dy}{dx}\right)-y=0$
4. $y{\left(\frac{dy}{dx}\right)}^2-2x\left(\frac{dy}{dx}\right)+y=0$

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. If $|z|=1$ and $|\omega -1|=1$ where $z,\omega \in C,$ then the largest set of values of $|2z-1{|}^2+|2\omega -1{|}^2$ equals

1. $[1,9]$
2. $[2,6]$
3. $[2,12]$
4. $[2,18]$

Q. The differential equations satisfied by the system of parabolas $y^2=4a(x+a)$ is:

1. $y{\left(\frac{dy}{dx}\right)}^2-2x\left(\frac{dy}{dx}\right)-y=0$
2. $y{\left(\frac{dy}{dx}\right)}^2+2x\left(\frac{dy}{dx}\right)-y=0$
3. $y\left(\frac{dy}{dx}\right)+2x\left(\frac{dy}{dx}\right)-y=0$
4. $y{\left(\frac{dy}{dx}\right)}^2-2x\left(\frac{dy}{dx}\right)+y=0$

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 circles $z\overline{z}+z\overline{a_1}+a_1\overline{z}+b_1=0,b_1\in R$ and $z\overline{z}+z\overline{a_2}+\overline{z_2}{a}_2+b_2=0,b_2\in R$ will intersect orthogonally if

1. $2\text{Im}\left(b_1\overline{b_2}\right)=a_1+a_2$
2. $2\text{Im}\left(\overline{b_1}{b}_2\right)=a_1+a_2$
3. $2\text{Re}\left(a_2\overline{a_1}\right)=b_1+b_2$
4. $2\text{Re}\left(a_1\overline{a_2}\right)=b_1+b_2$

Q. If $\text{Re}\left(\frac{z-1}{2z+i}\right)=1,$ where $z=x+iy$, then the point $(x,y)$ lies on a

1. circle whose centre is at $(-\frac{1}{2},-\frac{3}{2}).$
2. straight line whose slope is $\frac{3}{2}$.
3. circle whose diameter is $\frac{\sqrt{5}}{2}$.
4. straight line whose slope is $-\frac{2}{3}$.

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 '$z$' lies on the circle $|z-2i|=2\sqrt{2}$, then the value of $\arg \left[\frac{(z-2)}{(z+2)}\right]$ is equal to

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

Q. Which of the following option(s) is/are correct for circle represented by $\overline{z}=\overline{a}+\frac{r^2}{z-a}?$

1. Circle having centre at $-a$
2. Circle having centre at $a$
3. Radius of the circle $=r$
4. Radius of the circle $=r^2$

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 )