Question

The number of values of \(z \in \mathbb{C}\) satisfying \(|z-4-8i| = \sqrt{10}\) and \(|z-3-5i| + |z-5-11i| = 4\sqrt{5}\) is

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The Correct Option is B

Step 1: Identifying the Curves:
The first condition $|z - 4 - 8i| = \sqrt{10}$ represents a circle with center $C_1 = (4, 8)$ and radius $r = \sqrt{10}$.
The second condition $|z-3-5i| + |z-5-11i| = 4\sqrt{5}$ represents an ellipse with foci $F_1 = (3,5)$ and $F_2 = (5,11)$, and sum of distances $2a = 4\sqrt{5}$.
\includegraphics[width=0.5\linewidth]{m11.png}
Step 2: Computing Ellipse Parameters:
Distance between foci: \[ 2c = |F_1 F_2| = \sqrt{(5-3)^2 + (11-5)^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10} \] \[ c = \sqrt{10},\quad a = 2\sqrt{5} \] \[ b^2 = a^2 - c^2 = 20 - 10 = 10 \implies b = \sqrt{10} \] Step 3: Key Observation -- Circle Touches Ellipse:
The center of the circle $C_1 = (4,8)$ is the midpoint of the segment $F_1F_2$: \[ \text{Midpoint of }F_1F_2 = \left(\frac{3+5}{2}, \frac{5+11}{2}\right) = (4, 8) = C_1 \] So the circle is centered at the center of the ellipse.
The radius of the circle is $r = \sqrt{10} = b$ (the semi-minor axis of the ellipse).
This means the circle has radius equal to the semi-minor axis of the ellipse, and is centered at the center of the ellipse. Therefore the circle touches the ellipse internally at the two ends of the minor axis.
Step 4: Counting Intersection Points:
Since the circle internally touches the ellipse at exactly the two endpoints of the minor axis, there are exactly 2 intersection points.
Step 5: Final Answer:
The number of values of $z$ satisfying both conditions is $\mathbf{2}$.
The answer is Option (3).