Equation of Conics in Complex Form

Q. If $x^2+9y^2-4x+3=0,x,y\in R,$ then $x$ and $y$ respectively lie in the intervals

1. $[1,3]$ and $[1,3]$
2. $[-\frac{1}{3},\frac{1}{3}]$ and $[-\frac{1}{3},\frac{1}{3}]$
3. $[-\frac{1}{3},\frac{1}{3}]$ and $[1,3]$
4. $[1,3]$ and $[-\frac{1}{3},\frac{1}{3}]$

Q. The locus of point $z$ satisfying $Re\left(\frac{1}{z}\right)=k$, where $k$ is a non-zero real number is

1. a straight line
2. a circle
3. an ellipse
4. a hyperbola

Q. If $|z-2-3i{|}^2+|z-4-3i{|}^2=\lambda$ represents a equation of circle, then the value of $\lambda$ when the radius of circle is minimum, is

Q. Consider the ellipse $\frac{x^2}{f(k^2+3k+5)}+\frac{y^2}{f(k+13)}=1$. If $f\left(x\right)$ is positive decreasing function, then

1. Number of positive integral values of $k$ for which $x$ axis is major axis equals $1$
2. the set of values of $k$ for which t$y-$axis is the major axis is $(-\infty ,2)$
3. the set of values of $k$ for which $y-$axis is the major axis is $(-\infty ,-4)\cup (2,\infty )$
4. the set of values of $k$ for which $y-$axis is the major axis is $(-4,\infty )$

Q. The possible integral values of $k$ for which the equation
$|z+i|-|z-i|=k,k>0$ represents a hyperbola is

Q. If $z$ is a complex number, not purely real such that imaginary part of $z-1+\frac{1}{z-1}$ is zero, then locus of $z$ is

1. a circle of radius $1$ unit
2. a straight line parallel to $x$-axis
3. a hyperbola
4. a parabola with axis of symmetry parallel to $x$-axis

Q. If the maximum possible principal argument of the complex number $z$ satisfying $|z-4|=Re\left(z\right)$ is​​​​​​ $k$, then the value of $\frac{\pi }{k}$ is

Q. If $z$ is any complex number satisfying $|z-3-2i|\le 2,$ then the minimum value of $|2z-6+5i|$ is

Q. If $z$ is a complex number that satisfies $z^2+z|z|+|z{|}^2=0,$ then the locus of $z$ is

1. a straight line
2. a circle
3. a pair of straight lines
4. an ellipse

Q. Let $y=f\left(x\right)$ be a parabola having $(0,\frac{3}{2})$ and $(0,0)$ as vertex and focus respectively. Then the number of roots of the equation $\frac{1}{2f\left(x\right)-3}-\frac{1}{2f\left(x\right)+3}+\frac{6}{x^2}=0$ is

Q. Consider an elipse having its foci at $A\left(z_1\right)\text{and}B\left(z_2\right)$ in the argand plane. If the eccentricity of the ellipse is $e$ and it is known that origin is an interior point of the ellipse, then $e$ lies in the interval

1. $(0,\frac{|z_1-z_2|}{|z_1|+|z_2|})$
2. $(0,\frac{|z_1-z_2|}{|z_1^2|+|z_2{|}^2})$
3. $(0,\frac{|z_1|}{|z_2|})$
4. $(0,\frac{|z_2|}{|z_1|})$

Q. If $z=x+iy$ and $|z-1{|}^2+|z+1{|}^2=4$, then the locus of $z$ is

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

Q. If the equation $|z-a|+|z-b|=3$ represents an ellipse and $a,b\in R$, (where $a$ is fixed point), then $b$ lies

1. On the circumference of the circle $|z-a|=3$
2. None of these
3. Outside the circle $|z-a|=3$
4. Inside the circle $|z-a|=3$

Q. If $|z^2-1|=|z{|}^2+1$, then $z$ lies on

1. real axis
2. line $x=y$
3. line $x=2y$
4. imaginary axis

Q. The shortest distance between line $y-x=1$ and curve $x=y^2$ is

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

Q. Let $z_1$ satisfy the condition $|z-3|=2$ and $z_2$ satisfy the equation $|z-1|+|z+1|=3$. If $|z_1-z_2{|}_{min}=m$ and $|z_1-z_2{|}_{max}=M$, then which of the following is/are correct?

1. $M+m=\frac{9}{2}$
2. $M^2+m^2+Mm=\frac{81}{4}$
3. $M^2+m^2+Mm=\frac{169}{4}$
4. $M+m=\frac{13}{2}$

Q. If $Z=x+iy$ and ${}^{\prime }{a}^{\prime }$ is a real number such that $|z-ai|=|z+ai|$, then locus of $z$ is

1. x-axis
2. y-axis
3. $x^2+y^2=1$
4. $x=y$

Q. If $|z_1|=2,\&z_2$be the point in which sartisfies the condition $(z_2+\overline{z_2})-i(z_2-\overline{z_2})=8\sqrt{2},$ then the minimum value of $|z_1-z_2|$ is

Q. If $(a,b)$ is at unit distance from the line $8x+6y+1=0$, then which of the following conditions are correct?
1. $3a-4b-4=0$
2. $8a+6b+11=0$
3. $8a+6b-9=0$
Select the correct answer using the code given below:

1. 1 and 2 only
2. 2 and 3 only
3. 1 and 3 only
4. 1, 2 and 3

Q. If $|z-2-3i{|}^2+|z-4-3i{|}^2=\lambda$ represents a equation of circle, then the value of $\lambda$ when the radius of circle is minimum, is

Q. If $z$ is a complex number that satisfies $z^2+z|z|+|z{|}^2=0,$ then the locus of $z$ is

1. a circle
2. a straight line
3. a pair of straight lines
4. an ellipse

Q. Let $f\left(x\right)=x^3-3x^2-9x+9,$ then which of the following(s) is(are) correct

1. $f\left(x\right)$ has a local minima at $x=3$
2. $f\left(x\right)$ has a local maxima at $x=-1$
3. $f\left(x\right)$ has a local maxima at $x=3$
4. $f\left(x\right)$ has a local minima at $x=-1$

Q. The number of possible integral value(s) of $k$ for which the equation
$|z+i|-|z-i|=k,k>0$ represents a hyperbola is

Q. If the maximum possible principal argument of the complex number $z$ satisfying $|z-4|=Re\left(z\right)$ is​​​​​​ $k$, then the value of $\frac{\pi }{k}$ is

Q. If $|z-2-3i{|}^2+|z-4-3i{|}^2=\lambda$ represents a equation of circle, then the value of $\lambda$ when the radius of circle is minimum, is

Q. $\text{If}|z_1|=2,\&z_2\text{be the point in which sartisfies}$ $\text{the condition}(z_2+\overline{z_2})-i(z_2-\overline{z_2})=8\sqrt{2},$ then the minimum value of $|z_1-z_2|$ is

Q. If the maximum possible principal argument of the complex number $z$ satisfying $|z-4|=Re\left(z\right)$ is​​​​​​ $k$, then the value of $\frac{\pi }{k}$ is

Q. If $z=x+iy$ and $|z-1{|}^2+|z+1{|}^2=4$, then the locus of $z$ is

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

Q. If $|z^2-1|=|z{|}^2+1$, then $z$ lies on

1. imaginary axis
2. real axis
3. line $x=y$
4. line $x=2y$

Q. Let $f\left(x\right)=x^3-3x^2-9x+9,$ then which of the following(s) is(are) correct

1. $f\left(x\right)$ has a local minima at $x=3$
2. $f\left(x\right)$ has a local maxima at $x=-1$
3. $f\left(x\right)$ has a local minima at $x=-1$
4. $f\left(x\right)$ has a local maxima at $x=3$