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

To find the maximum distance between the two curves given by the equations:

\(|z-2| = 4\)

and

\(|z-2| + |z+2| = 5\)

1. We start by understanding the geometry of the given equations. The equation \(|z-2| = 4\) represents a circle centered at \((2, 0)\) with radius \(4\).
2. The equation \(|z-2| + |z+2| = 5\) represents an ellipse with foci at \((2,0)\) and \((-2,0)\), with the sum of the distances to the foci equal to \(5\).

Next, we calculate the maximum distance between these two curves.

1. For the circle \(|z-2|=4\), the extreme right point on the circle is \((6,0)\), and the extreme left point is \((-2,0)\).
2. For the ellipse, we find the intersection of the ellipse with the x-axis. The ellipse in its simplest co-ordinate representation, considering its foci is:

\frac{|x-2|}{5-2} + \frac{|x+2|}{5-2} = 1

1. By solving the above linear equation, we get the extreme values for x on the ellipse as \(1.5\) and \(-1.5\). Therefore, the maximum x-values for the ellipse are \((2.5, 0)\) and \((-2.5, 0)\).
2. The maximum distance between a point on the circle and a point on the ellipse therefore occurs between the extreme right points of both:

The distance:

\text{Distance} = \text{Rightmost point of circle} - \text{leftmost point of ellipse} = 6 - (-2.5) = 8.5

 

Thus, the maximum distance between the given curves is \(\dfrac{17}{2}\).

Hence, the correct answer is \(\dfrac{17}{2}\).