Let Z₁ and Z₂ be two complex numbers satisfying |Z₁| = 9 and |Z₂ - 3 - 4i| = 4. Then the minimum val

Q: Wait, why is $|Z - Z_0| = r$ a circle? I thought complex numbers were just points.

That is a brilliant observation! In the Argand plane, a complex number $Z$ is indeed a point $(x, y)$. The expression $|Z - Z_0|$ represents the distance between the point $Z$ and the fixed point $Z_0$. So, $|Z - Z_0| = r$ is simply the locus of all points $Z$ that are at a constant distance $r$ from $Z_0$. That is the definition of a circle!

Q: How do I know if the circles touch internally or externally?

Great question. It all comes down to the relationship between the distance between centers $d$ and the radii $r_1$ and $r_2$. If $d = r_1 + r_2$, they touch externally. If $d = |r_1 - r_2|$, they touch internally. In our case, $d = 5$ and $|9 - 4| = 5$, so they must be touching internally.

Q: Why is the minimum distance zero? Shouldn't it be something else?

If the circles were separate, the minimum distance would be $d - r_1 - r_2$. But because they touch internally, they share a common point. At that specific point, $Z_1$ and $Z_2$ can be identical. Since the distance between a point and itself is zero, the minimum possible distance is indeed zero.

Question

Let $Z_1$ and $Z_2$ be two complex numbers satisfying $|Z_1| = 9$ and $|Z_2 - 3 - 4i| = 4$. Then the minimum value of $|Z_1 - Z_2|$ is :

Options:

(A) $0$

(B) $1$

(C) $\sqrt{2}$

(D) $2$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

My dear student, today we are going to embark on a journey that transforms the abstract world of complex numbers into the tangible, beautiful realm of geometry. Often, when we see equations like $|Z_1| = 9$ and $|Z_2 - 3 - 4i| = 4$, our instinct is to dive into algebraic manipulation, perhaps substituting $Z = x + iy$.

But stop! Take a breath. Let us look at these equations with the eyes of a geometer.

### The Argand Plane

Imagine you are standing on the Argand plane. The complex number $Z_1$ is a point, and the condition $|Z_1| = 9$ tells us that the distance from the origin $(0, 0)$ to $Z_1$ is always $9$. This is the definition of a circle centered at the origin with a radius $r_1 = 9$.

Now, look at the second condition: $|Z_2 - (3 + 4i)| = 4$. This is just a circle shifted from the origin. The center is $C_2 = (3, 4)$ and the radius is $r_2 = 4$. We have two circles, and we want to minimize the distance between a point on the first circle and a point on the second.

### The Distance Between Centers

To understand how these circles interact, we must find the distance between their centers. Our first center is $C_1(0, 0)$ and our second is $C_2(3, 4)$.

Using the distance formula, we calculate the distance $d$ as follows:

$$d = \sqrt{(3 - 0)^2 + (4 - 0)^2}$$

This simplifies beautifully:

$$d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$

The distance between the centers is exactly $5$.

### The Moment of Revelation

Now, let us compare this distance $d = 5$ with the radii of our circles. We have $r_1 = 9$ and $r_2 = 4$.

Notice that the difference between the radii is:

$$|r_1 - r_2| = |9 - 4| = 5$$

We have discovered that $d = |r_1 - r_2|$. In the world of geometry, this is a special condition. It means the two circles are touching each other internally. One circle is nestled perfectly inside the other, kissing at exactly one point.

### The Final Conclusion

Since the circles touch at a single point, there exists a complex number that lies on both circles simultaneously. If $Z_1$ and $Z_2$ are both at this point of contact, then $Z_1 = Z_2$.

The distance $|Z_1 - Z_2|$ becomes $|Z_1 - Z_1| = 0$. Thus, the **minimum value of the distance is $0$**.

See how the complexity vanished once we visualized the geometry? Keep this perspective, and no complex number problem will ever intimidate you again.

Animated Solution for Mathematics - Complex Numbers: Let $Z_{1}$ and $Z_{2}$ be two complex numbers satisfying $∣ Z_{1} ∣ = 9$ and $∣ Z_{2} - 3 - 4 i ∣ = 4$ . Then the minimum value of $∣ Z_{1} - Z_{2} ∣$ is :

Select Answer:

Visualized Solution

The Argand Plane

Standard Circle Equation

Locus of $Z_{1}$

Locus of $Z_{2}$

Distance Formula

Substituting Center Coordinates

Calculating Center Distance

Difference of Radii

Condition for Internal Touching

Minimum Distance $∣ Z_{1} - Z_{2} ∣$

The Sigma Insight: Geometrical Applications of Complex Numbers

Analyzing the Setup

The Argand Plane

The Distance Between Centers

The Moment of Revelation

The Final Conclusion

Similar Questions

For all complex numbers $z_{1}, z_{2}$ satisfying $∣ z_{1} ∣ = 12$ and $∣ z_{2} - 3 - 4 i ∣ = 5$ , the minimum value of $∣ z_{1} - z_{2} ∣$ is

Let $∣ z_{1} - 8 - 2 i ∣ \leq 1$ and $∣ z_{2} - 2 + 6 i ∣ \leq 2, z_{1}, z_{2} \in C$ . Then the minimum value of $∣ z_{1} - z_{2} ∣$ is :

If $z$ is a complex number such that $∣ z ∣ \geq 1$ , then the minimum value of $∣ z + \frac{1}{2} (3 + 4 i) ∣$ is:

If $z$ is a complex number such that $\frac{z - i}{z - 1}$ is purely imaginary, then the minimum value of $∣ z - (3 + 3 i) ∣$ is:

A point $z$ moves in the complex plane such that $ar g (\frac{z - 2}{z + 2}) = \frac{π}{4}$ , then the minimum value of $∣ z - 92 - 2 i ∣^{2}$ is equal to .

Let $S_{1} = {z_{1} \in C : ∣ z_{1} - 3∣ = \frac{1}{2}}$ and $S_{2} = {z_{2} \in C : ∣ z_{2} - ∣ z_{2} + 1∣∣ = ∣ z_{2} + ∣ z_{2} - 1∣∣}$ . Then, for $z_{1} \in S_{1}$ and $z_{2} \in S_{2}$ , the least value of $∣ z_{2} - z_{1} ∣$ is:

If $∣ z - 3 + 2 i ∣ \leq 4$ then the difference between the greatest value and the least value of $∣ z ∣$ is :-

If $z$ is any complex number satisfying $∣ z - 3 - 2 i ∣ \leq 2$ , then the minimum value of $∣2 z - 6 + 5 i ∣$ is .........

Let $z_{1} = 10 + 6 i$ and $z_{2} = 4 + 6 i$ . If $z$ is any complex number such that the argument of $\frac{z - z _{1}}{z - z _{2}}$ is $\frac{π}{4}$ , then prove that $∣ z - 7 - 9 i ∣ = 32$ .

For $z \in C$ if the minimum value of $(∣ z - 32 ∣ + ∣ z - p 2 i ∣)$ is $52$ , then a value of $p$ is

Select Answer:

Visualized Solution

The Argand Plane

Standard Circle Equation

Locus of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Z1​

Locus of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Z2​

Distance Formula

Substituting Center Coordinates

Calculating Center Distance

Difference of Radii

Condition for Internal Touching

Minimum Distance .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∣Z1​−Z2​∣

The Sigma Insight: Geometrical Applications of Complex Numbers

Analyzing the Setup

The Argand Plane

The Distance Between Centers

The Moment of Revelation

The Final Conclusion

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For all complex numbers z1​,z2​ satisfying ∣z1​∣=12 and ∣z2​−3−4i∣=5, the minimum value of ∣z1​−z2​∣ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let ∣z1​−8−2i∣≤1 and ∣z2​−2+6i∣≤2,z1​,z2​∈C. Then the minimum value of ∣z1​−z2​∣ is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If z is a complex number such that ∣z∣≥1, then the minimum value of ∣z+21​(3+4i)∣ is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If z is a complex number such that z−1z−i​ is purely imaginary, then the minimum value of ∣z−(3+3i)∣ is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A point z moves in the complex plane such that arg(z+2z−2​)=4π​, then the minimum value of ∣z−92​−2i∣2 is equal to .

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ∣z−3+2i∣≤4 then the difference between the greatest value and the least value of ∣z∣ is :-

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If z is any complex number satisfying ∣z−3−2i∣≤2, then the minimum value of ∣2z−6+5i∣ is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z1​=10+6i and z2​=4+6i. If z is any complex number such that the argument of z−z2​z−z1​​ is 4π​, then prove that ∣z−7−9i∣=32​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For z∈C if the minimum value of (∣z−32​∣+∣z−p2​i∣) is 52​, then a value of p is

Locus of $Z_{1}$

Locus of $Z_{2}$

Minimum Distance $∣ Z_{1} - Z_{2} ∣$

For all complex numbers $z_{1}, z_{2}$ satisfying $∣ z_{1} ∣ = 12$ and $∣ z_{2} - 3 - 4 i ∣ = 5$ , the minimum value of $∣ z_{1} - z_{2} ∣$ is

Let $∣ z_{1} - 8 - 2 i ∣ \leq 1$ and $∣ z_{2} - 2 + 6 i ∣ \leq 2, z_{1}, z_{2} \in C$ . Then the minimum value of $∣ z_{1} - z_{2} ∣$ is :

If $z$ is a complex number such that $∣ z ∣ \geq 1$ , then the minimum value of $∣ z + \frac{1}{2} (3 + 4 i) ∣$ is:

If $z$ is a complex number such that $\frac{z - i}{z - 1}$ is purely imaginary, then the minimum value of $∣ z - (3 + 3 i) ∣$ is:

A point $z$ moves in the complex plane such that $ar g (\frac{z - 2}{z + 2}) = \frac{π}{4}$ , then the minimum value of $∣ z - 92 - 2 i ∣^{2}$ is equal to .

If $∣ z - 3 + 2 i ∣ \leq 4$ then the difference between the greatest value and the least value of $∣ z ∣$ is :-

If $z$ is any complex number satisfying $∣ z - 3 - 2 i ∣ \leq 2$ , then the minimum value of $∣2 z - 6 + 5 i ∣$ is .........

Let $z_{1} = 10 + 6 i$ and $z_{2} = 4 + 6 i$ . If $z$ is any complex number such that the argument of $\frac{z - z _{1}}{z - z _{2}}$ is $\frac{π}{4}$ , then prove that $∣ z - 7 - 9 i ∣ = 32$ .

For $z \in C$ if the minimum value of $(∣ z - 32 ∣ + ∣ z - p 2 i ∣)$ is $52$ , then a value of $p$ is