Question

If the complex number z lies on a circle with center at origin and radius $\dfrac{1}{4}$, then the complex number -1+8z lies on a circle with radius
\[\begin{align}
  & A.4 \\
 & B.1 \\
 & C.3 \\
 & D.2 \\
\end{align}\]

Answer 1

Hint: In this question, we are given the radius and center of a circle on which complex number z lies. We have to determine the radius of the circle on which -1+8z lies. For this, we will first form an equation of the given circle using the given center and radius. And then use it to represent -1+8z in the form of an equation of the circle which will give us the required radius. For any complex number z lying on the circle with radius r and center c, the equation is given by $\left| z-c \right|=r$.

Complete step-by-step solution
Here, we are given the radius of the circle to be $\dfrac{1}{4}$ and the center of the circle to be at origin i.e. (0,0). Since complex number z lies on this circle, therefore, equation of circle can be written as
\[\begin{align}
  & \left| z-0 \right|=\dfrac{1}{4} \\
 & \Rightarrow \left| z \right|=\dfrac{1}{4}\cdots \cdots \cdots \cdots \left( 1 \right) \\
\end{align}\]
Now, we need to find an equation of the circle on which -1+8z lies. For this, first let us suppose that,
\[\Rightarrow z'=-1+8z\]
Take 1 on other side we get:
\[\Rightarrow z'+1=8z\]
Dividing by 8 both sides we get:
\[\Rightarrow \dfrac{z'+1}{8}=z\]
Now let us take magnitude (modulus) both side, we get:
\[\Rightarrow \left| \dfrac{z'+1}{8} \right|=\left| z \right|\]
As we know, $\left| \dfrac{x}{y} \right|=\dfrac{\left| x \right|}{\left| y \right|}\text{ and }\left| a \right|=a$ where 'a' is constant, so our equation becomes,
\[\Rightarrow \left| \dfrac{z'+1}{8} \right|=\left| z \right|\]
Putting value of $\left| z \right|$ from (1), we get:
\[\Rightarrow \left| \dfrac{z'+1}{8} \right|=\dfrac{1}{4}\]
Taking 8 on other side, we get:
\[\begin{align}
  & \Rightarrow \left| z'+1 \right|=\dfrac{8}{4} \\
 & \Rightarrow \left| z'+1 \right|=2 \\
\end{align}\]
Hence, this is an equation of circle on which z'=-1+8z lies.
Comparing it with $\left| z-c \right|=r$ we get: c = -1 and r = 2, where c represents center and r represents radius of circle.
Therefore, 2 is the radius of the circle on which -1+8z lies.
Hence, option D is the correct answer.

Note: Students should know how to form an equation in complex form. While comparing any equation with $\left| z-c \right|=r$ take care of the sign of c. We have taken $\left| z+1 \right|$ as $\left| z-\left( -1 \right) \right|$ for comparing it to $\left| z-c \right|$ to get c = -1. Students can get confused between c and r. Here, $\left| z-c \right|=r$ represents that z is a point which is at distance r from center and thus as z moves, it will form a circle.