Key Concepts and Formulas

• Polar Form of Complex Numbers: A complex number $z$ can be represented as $z = re^{i\theta}$, where $r = |z|$ is the modulus and $\theta = \arg(z)$ is the argument.
• Euler's Formula: $e^{i\theta} = \cos\theta + i\sin\theta$.
• Equation of an Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a$ is the semi-major axis and $b$ is the semi-minor axis.

Step-by-Step Solution

Step 1: Understand Curve $C_1$

• What & Why: We need to interpret the equation $|z| = 4$. This represents all complex numbers $z$ whose distance from the origin is 4.
• Math: $|z| = 4$
• Reasoning: This is the definition of a circle centered at the origin with radius 4. We can represent $z$ in polar form as $z = 4e^{i\theta}$, where $\theta$ varies from $0$ to $2\pi$.

Step 2: Express $w$ in terms of $\theta$

• What & Why: We are given $w = z + \frac{1}{z}$ and we need to express $w$ in terms of the parameter $\theta$ from the polar form of $z$. This will allow us to find the locus of $w$.
• Math: $w = z + \frac{1}{z} = 4e^{i\theta} + \frac{1}{4e^{i\theta}} = 4e^{i\theta} + \frac{1}{4}e^{-i\theta}$
• Reasoning: We substitute the polar form of $z$ into the expression for $w$ and use the property that $\frac{1}{e^{i\theta}} = e^{-i\theta}$.

Step 3: Apply Euler's Formula

• What & Why: We use Euler's formula to express $e^{i\theta}$ and $e^{-i\theta}$ in terms of $\cos\theta$ and $\sin\theta$. This allows us to separate the real and imaginary parts of $w$.
• Math: $w = 4(\cos\theta + i\sin\theta) + \frac{1}{4}(\cos\theta - i\sin\theta)$
• Reasoning: We substitute $e^{i\theta} = \cos\theta + i\sin\theta$ and $e^{-i\theta} = \cos\theta - i\sin\theta$ into the expression for $w$.

Step 4: Separate Real and Imaginary Parts

• What & Why: We group the real and imaginary terms to express $w$ in the form $x + iy$, where $x$ and $y$ are functions of $\theta$.
• Math: $w = \left(4\cos\theta + \frac{1}{4}\cos\theta\right) + i\left(4\sin\theta - \frac{1}{4}\sin\theta\right) = \frac{17}{4}\cos\theta + i\frac{15}{4}\sin\theta$
• Reasoning: We combine the real and imaginary terms to get $w = x + iy$, where $x = \frac{17}{4}\cos\theta$ and $y = \frac{15}{4}\sin\theta$.

Step 5: Eliminate $\theta$ to find the Locus

• What & Why: We eliminate the parameter $\theta$ to find the Cartesian equation relating $x$ and $y$. This equation represents the locus of $w$.
• Math: $x = \frac{17}{4}\cos\theta \implies \cos\theta = \frac{4x}{17}$ $y = \frac{15}{4}\sin\theta \implies \sin\theta = \frac{4y}{15}$ Using $\cos^2\theta + \sin^2\theta = 1$: $\left(\frac{4x}{17}\right)^2 + \left(\frac{4y}{15}\right)^2 = 1$ $\frac{16x^2}{289} + \frac{16y^2}{225} = 1$ $\frac{x^2}{\left(\frac{17}{4}\right)^2} + \frac{y^2}{\left(\frac{15}{4}\right)^2} = 1$
• Reasoning: We use the trigonometric identity $\cos^2\theta + \sin^2\theta = 1$ to eliminate $\theta$ and obtain the equation of an ellipse.

Step 6: Analyze the Intersection of $C_1$ and $C_2$

• What & Why: We need to determine how the circle $C_1$ and the ellipse $C_2$ intersect. Since we know $C_1$ is a circle with radius 4 and $C_2$ is an ellipse with semi-major axis $a = \frac{17}{4} = 4.25$ and semi-minor axis $b = \frac{15}{4} = 3.75$, we can compare their dimensions.
• Math:
  • $C_1: x^2 + y^2 = 16$
  • $C_2: \frac{x^2}{(17/4)^2} + \frac{y^2}{(15/4)^2} = 1$
• Reasoning: Since $4.25 > 4$ and $3.75 < 4$, the ellipse extends beyond the circle along the x-axis and is contained within the circle along the y-axis. This implies that the two curves intersect. To find the number of intersection points, we solve for $x$ and $y$ values that satisfy both equations. Substituting $x^2 = 16-y^2$ into the ellipse equation: $\frac{16-y^2}{(17/4)^2} + \frac{y^2}{(15/4)^2} = 1$ $\frac{16(16-y^2)}{289} + \frac{16y^2}{225} = 1$ $225(16-y^2) + 289y^2 = \frac{289 \cdot 225}{16}$ $3600 - 225y^2 + 289y^2 = \frac{65025}{16}$ $64y^2 = \frac{65025}{16} - 3600 = \frac{65025 - 57600}{16} = \frac{7425}{16}$ $y^2 = \frac{7425}{64 \cdot 16} = \frac{7425}{1024}$ $y = \pm \sqrt{\frac{7425}{1024}} = \pm \frac{15\sqrt{33}}{32}$ Since we get two real values for y, and for each y, we get two values of x, there are four points of intersection.

Step 7: Verify the Number of Intersection Points (Alternative Method)

• What & Why: We already found that $\cos^2\theta = \frac{31}{64}$ and $\sin^2\theta = \frac{33}{64}$. We analyze the possible combinations of $\cos\theta$ and $\sin\theta$ to determine the number of intersection points.
• Math: $\cos\theta = \pm\frac{\sqrt{31}}{8} \quad \text{and} \quad \sin\theta = \pm\frac{\sqrt{33}}{8}$
• Reasoning: Since we have two possible values for $\cos\theta$ and two possible values for $\sin\theta$, we have four possible combinations. Each combination corresponds to a unique point on the complex plane. Therefore, there are four intersection points.

Common Mistakes & Tips

• Incorrectly applying Euler's Formula: Ensure you correctly substitute and simplify when using Euler's formula.
• Algebra Errors: Be careful with algebraic manipulations when eliminating the parameter $\theta$.
• Geometric Intuition: Sketching a quick diagram of the circle and ellipse can help you visualize the intersection.

Summary

We found that the locus of $w = z + \frac{1}{z}$ is an ellipse centered at the origin with semi-major axis $\frac{17}{4}$ and semi-minor axis $\frac{15}{4}$. By analyzing the dimensions of the circle $C_1$ and the ellipse $C_2$, and then verifying with trigonometric identities, we determined that the two curves intersect at four points.

Final Answer

The final answer is \boxed{4}, which corresponds to option (A).