Question

Find the maximum distance between the two curves: \[ |z-2| = 4 \quad \text{and} \quad |z-2| + |z+2| = 5 \]

• A. \(\dfrac{17}{2}\)
• B. \(\dfrac{15}{2}\)
• C. \(8\)
• D. \(9\)

Hint:

For distance between two loci, always check points lying on the {same line joining their centres or foci} — extrema occur there.

Answer 1

The Correct Option is A

Step 1: Interpret the loci


The equation \(|z-2| = 4\) represents a circle
with centre at \(z=2\) (i.e. point \((2,0)\)) and radius \(4\).
The equation \(|z-2| + |z+2| = 5\) represents an ellipse
with foci at \(z=2\) and \(z=-2\).
Distance between the foci: \[ 2c = 4 \Rightarrow c=2 \] For ellipse: \[ 2a = 5 \Rightarrow a = \frac{5}{2} \]
Step 2: Find semi-minor axis of ellipse
\[ b = \sqrt{a^2 - c^2} = \sqrt{\left(\frac{5}{2}\right)^2 - 2^2} = \sqrt{\frac{25}{4} - \frac{16}{4}} = \frac{3}{2} \] So the ellipse is centered at the origin with: \[ \text{major axis endpoints at } \pm \frac{5}{2} \]
Step 3: Determine farthest points


Farthest point on the circle from the origin lies at: \[ x = 2 + 4 = 6 \]
Farthest point on the ellipse from the origin lies at: \[ x = -\frac{5}{2} \] (opposite direction for maximum separation)

Step 4: Compute maximum distance
\[ \text{Maximum distance} = 6 - \left(-\frac{5}{2}\right) = \frac{12}{2} + \frac{5}{2} = \frac{17}{2} \] \[ \boxed{\dfrac{17}{2}} \]