For all `z in C` on the curve `C_(1):abs(z)=4`,let the locus of the point `z+(1)/(z)` be the curve `C_(2)`.Then :

Options

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

Answer

To solve the problem, we need to find the locus of the point \( w = z + \frac{1}{z} \) where \( z \) lies on the curve \( C_1: |z| = 4 \).

### Step-by-Step Solution:

1. **Express \( z \) in polar form**:
Since \( z \) lies on the circle of radius 4, we can express \( z \) in polar form as:
\[ z = 4 e^{i\theta} \]
where \( \theta \) is the argument of \( z \).

2. **Substitute \( z \) into \( w \)**:
Now, substitute \( z \) into the expression for \( w \):
\[ w = z + \frac{1}{z} = 4 e^{i\theta} + \frac{1}{4 e^{i\theta}} \]
Simplifying the second term:
\[ w = 4 e^{i\theta} + \frac{1}{4} e^{-i\theta} \]

3. **Combine the terms**:
We can combine the two terms:
\[ w = 4 e^{i\theta} + \frac{1}{4} e^{-i\theta} = 4 \left( \cos \theta + i \sin \theta \right) + \frac{1}{4} \left( \cos \theta - i \sin \theta \right) \]
This simplifies to:
\[ w = \left( 4 + \frac{1}{4} \right) \cos \theta + i \left( 4 - \frac{1}{4} \right) \sin \theta \]
\[ w = \frac{17}{4} \cos \theta + i \frac{15}{4} \sin \theta \]

4. **Identify the locus**:
Let \( x = \frac{17}{4} \cos \theta \) and \( y = \frac{15}{4} \sin \theta \). To find the relationship between \( x \) and \( y \), we can use the identity \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[ \left( \frac{4x}{17} \right)^2 + \left( \frac{4y}{15} \right)^2 = 1 \]
Simplifying this gives:
\[ \frac{x^2}{\left( \frac{17}{4} \right)^2} + \frac{y^2}{\left( \frac{15}{4} \right)^2} = 1 \]

5. **Final equation of the locus**:
The equation of the locus is:
\[ \frac{x^2}{\frac{289}{16}} + \frac{y^2}{\frac{225}{16}} = 1 \]
Multiplying through by 16 gives:
\[ \frac{16x^2}{289} + \frac{16y^2}{225} = 1 \]

### Conclusion:
The locus of the point \( w = z + \frac{1}{z} \) is an ellipse centered at the origin.