The locus of the centre of a circle which touches the circle |z-z1| = a and |z-z2| = b externally (z

Q: Wait, why is it a hyperbola and not an ellipse? Isn't the locus of a point with a constant sum of distances an ellipse?

That is a brilliant observation! You are absolutely correct that the locus of a point where the *sum* of distances to two fixed points is constant is an ellipse. However, in this problem, we are looking at the *difference* of distances. When we subtract the two equations, the variable radius $r$ cancels out, leaving us with $|z - z_1| - |z - z_2| = a - b$. Since $a$ and $b$ are fixed, $a - b$ is a constant. The locus of a point whose distance difference from two fixed points is constant is the definition of a hyperbola.

Q: What happens if the radii $a$ and $b$ are equal? Does the hyperbola still exist?

That is a fantastic edge case to consider! If $a = b$, then $a - b = 0$. The equation becomes $|z - z_1| = |z - z_2|$. Geometrically, this means the point $z$ is equidistant from $z_1$ and $z_2$. The locus then becomes the perpendicular bisector of the line segment joining $z_1$ and $z_2$. In the context of conic sections, this is a degenerate hyperbola.

Q: Why do we assume external touch? Does internal touch change the answer?

Great question. If the circles touched internally, the distance between their centers would be $|r - a|$ or $|a - r|$ instead of $r + a$. This would change the constant term and potentially the nature of the locus. Always read the problem carefully—'externally' is the key word here that defines our path. In JEE, the distinction between internal and external contact is a classic trap!

Question

The locus of the centre of a circle which touches the circle $|z-z_1| = a$ and $|z-z_2| = b$ externally ($z, z_1$ & $z_2$ are complex numbers) will be

Options:

(A) an ellipse

(B) a hyperbola

(C) a circle

(D) none of these

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineer! Today, we are going to embark on a journey through the complex plane. Imagine you are standing on a vast, infinite sheet of paper—the Argand plane.

On this plane, we have two fixed anchors: two circles, $C_1$ and $C_2$, with centers at $z_1$ and $z_2$ and radii $a$ and $b$, respectively. These circles are our static foundation.

Now, imagine a third, smaller circle—a wanderer—moving across this plane. Its center is at $z$, and its radius $r$ is constantly changing. Our mission is to uncover the secret path, the locus, that this center $z$ traces as it moves.

### The External Touch Condition

Let us look at the physics of the situation. The problem states that our moving circle touches both fixed circles externally.

If you were to draw a line connecting their centers, that line would pass through the point of contact. The distance between the centers is simply the sum of their radii.

For the first circle, the distance between the center of the moving circle $z$ and the center $z_1$ is $|z - z_1|$. Since they touch externally, we have the elegant equation:
$$|z - z_1| = r + a$$

Similarly, for the second circle, the distance between $z$ and $z_2$ is $|z - z_2|$, giving us:
$$|z - z_2| = r + b$$

### The Algebraic Dance

Now, we face a challenge. We want the locus of $z$, but our equations are cluttered with the variable $r$. We need to eliminate it.

We have two equations:
1) $|z - z_1| = r + a$
2) $|z - z_2| = r + b$

If we subtract the second equation from the first, the $r$ and $-r$ terms cancel each other out perfectly. We are left with:
$$|z - z_1| - |z - z_2| = a - b$$

This is a moment of pure mathematical satisfaction. The variable radius, which seemed to complicate everything, has vanished, leaving behind a relationship that depends only on the fixed parameters of our system.

### The Hyperbola Reveal

Look closely at the result: $|z - z_1| - |z - z_2| = a - b$. Since $a$ and $b$ are fixed radii, their difference $a - b$ is a constant.

Let us call this constant $k$. So, we have:
$$|z - z_1| - |z - z_2| = k$$

This is the classic geometric definition of a **hyperbola**. A hyperbola is defined as the locus of a point such that the absolute difference of its distances from two fixed points (the foci) is constant.

Here, our fixed points $z_1$ and $z_2$ are the foci of the hyperbola. The center of our moving circle $z$ is tracing out this beautiful, sweeping curve.

You have successfully navigated the complexity and arrived at the elegant truth: the locus is a **hyperbola**. Keep this intuition with you—whenever you see a difference of distances to fixed points, think hyperbola; whenever you see a sum, think ellipse.

Animated Solution for Mathematics - Complex Numbers: The locus of the centre of a circle which touches the circle $∣ z - z_{1} ∣ = a$ and $∣ z - z_{2} ∣ = b$ externally ( $z, z_{1}$ & $z_{2}$ are complex numbers) will be

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Visualized Solution

The Fixed Circles

The Moving Circle

External Touch Condition for $C_{1}$

Equation for First Circle

External Touch Condition for $C_{2}$

Equation for Second Circle

Eliminating the Variable Radius

Subtracting the Equations

Simplifying the Difference

Analyzing the Constant Difference

Definition of a Hyperbola

Final Conclusion

The Sigma Insight: Geometrical Applications of Complex Numbers

Analyzing the Setup

The External Touch Condition

The Algebraic Dance

The Hyperbola Reveal

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