Question

For all $z \in C$ on the curve $C_1:|z|=4$, let the locus of the point $z+\frac{1}{z}$ be the curve $C_2$ Then:

• A. the curve $C_1$ lies inside $C_2$
• B. the curves $C_1$ and $C_2$ intersect at 4 points
• C. the curve $C_2$ lies inside $C_1$
• D. the curves $C_1$ and $C_2$ intersect at 2 points

Hint:

For locus problems involving complex numbers, carefully use the geometric interpretations of the given relations to analyze the intersections of the curves.

Answer 1

The Correct Option is B

Let \(w=z+\frac{1}{z}​=4e^{iθ}+\frac{1}{4}​e^{−iθ}\) 

\(⇒w=\frac{17}{4}​\;cos\;θ+i\frac{15}{4}​\;sin\;θ \)

So locus of w is ellipse \(\frac{x^2}{(\frac{17}{4})^2}+\frac{y^2}{(\frac{15}{4})^2}=1\)

Locus of \(z\) is circle \(x^2+y^2=16\) 

So intersect at \(4\) points

The Correct Option is (B): the curves $C_1$ and $C_2$ intersect at 4 points

Answer 2

Step 1: We are given that \( C_1 \) is the curve of points \( z + \frac{1}{z} \). Let \( z = e^{i\theta} \) (since \( |z| = 1 \)), then

\[ z + \frac{1}{z} = e^{i\theta} + e^{-i\theta} = 2 \cos \theta \]

Step 2: So, the locus of \( z + \frac{1}{z} \) is a line. This is a key result, and we see that the expression simplifies to a trigonometric function, showing that the locus is a straight line determined by \( \theta \). This helps us visualize the behavior of the function over different values of \( \theta \), and the geometry becomes easier to interpret.

\[ \text{Locus of } z + \frac{1}{z} = 2 \cos \theta \]

Step 3: Using the given relations, the two curves \( C_1 \) and \( C_2 \) intersect at 4 points. This suggests that there are specific values of \( \theta \) where the curves cross, which can be found by solving for the points of intersection. These intersections give us valuable information about the relationship between the two curves. Solving for these intersection points reveals specific values of \( \theta \), which further helps in understanding the dynamic behavior of the curves in the complex plane.