For z ∈ C if the minimum value of (|z - 3√2| + |z - p√2i|) is 5√2, then a value of p is

Q: Wait, why can't we just use calculus to minimize this function?

You absolutely could! You would express $z$ as $x + iy$, substitute it into the modulus expression, and then minimize the function $f(x, y) = \sqrt{(x-3\sqrt{2})^2 + y^2} + \sqrt{x^2 + (y-p\sqrt{2})^2}$. However, this leads to a very tedious algebraic mess. The geometric approach using the Triangle Inequality is the 'JEE way'—it reveals the underlying structure of the problem and saves you precious time.

Q: Why is the minimum value exactly the distance between the two points?

Think of the Triangle Inequality: $|z - z_1| + |z - z_2| \ge |z_1 - z_2|$. The sum of two sides of a triangle is always greater than or equal to the third side. The equality holds if and only if the point $z$ is collinear with $z_1$ and $z_2$, specifically lying on the segment between them. In that configuration, the 'triangle' collapses into a straight line, and the sum of the distances is just the length of that line segment.

Q: I got $p = \pm 4$, but the options only show $4$. Did I do something wrong?

Not at all! You solved it perfectly. The question asks for 'a' value of $p$, implying that there might be multiple valid solutions. Since $4$ is one of the valid solutions and it appears in your options, you have successfully identified the correct answer. Always trust your derivation!

Question

For $z \in \mathbb{C}$ if the minimum value of $(|z - 3\sqrt{2}| + |z - p\sqrt{2}i|)$ is $5\sqrt{2}$, then a value of $p$ is

Options:

(A) 3

(B) 7/2

(C) 4

(D) 9/2

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, fellow traveler of the mathematical landscape. Today, we are not just solving a problem; we are exploring the elegant geometry hidden within the complex plane.

Imagine the Argand plane as your canvas. We are given the expression $|z - 3\sqrt{2}| + |z - p\sqrt{2}i|$ and told that its minimum value is $5\sqrt{2}$.

### Translating Algebra into Geometry

Look closely at the terms inside the modulus. In the complex plane, $|z - z_0|$ represents the distance between the variable point $z$ and a fixed point $z_0$.

By defining $z_1 = 3\sqrt{2}$ and $z_2 = p\sqrt{2}i$, our expression transforms into the sum of the distances from $z$ to these two fixed points. We are essentially asking: "If I have two fixed stakes in the ground at $z_1$ and $z_2$, where should I stand so that the total distance to both is minimized?"

### The Power of the Triangle Inequality

This is where the magic happens. The Triangle Inequality tells us that for any point $z$, the sum of the distances to two fixed points $z_1$ and $z_2$ is always greater than or equal to the direct distance between $z_1$ and $z_2$.

Mathematically, this is expressed as:
$$|z - z_1| + |z - z_2| \ge |z_1 - z_2|$$

The minimum occurs when $z$ lies directly on the line segment connecting $z_1$ and $z_2$. In this state of perfect alignment, the sum of the distances is exactly equal to the distance between the two fixed points.

### The Engine Room: Calculation

Now that we have identified the geometric condition, the rest is a beautiful, straightforward calculation. We are given that the minimum value is $5\sqrt{2}$.

Therefore, we set the distance between our fixed points equal to this value:
$$|z_1 - z_2| = 5\sqrt{2}$$

Substituting our values, we get $|3\sqrt{2} - p\sqrt{2}i| = 5\sqrt{2}$. To find the magnitude of this complex number, we use the formula $\sqrt{a^2 + b^2}$, where $a = 3\sqrt{2}$ and $b = -p\sqrt{2}$.

This gives us:
$$\sqrt{(3\sqrt{2})^2 + (-p\sqrt{2})^2} = 5\sqrt{2}$$

Squaring both sides to eliminate the radical, we obtain:
$$(3\sqrt{2})^2 + (-p\sqrt{2})^2 = (5\sqrt{2})^2$$

Expanding these squares, we find:
$$18 + 2p^2 = 50$$

Subtracting $18$ from both sides yields $2p^2 = 32$, which simplifies to $p^2 = 16$.

Thus, **$p = \pm 4$**. Since $4$ is one of our options, we have arrived at our destination.

Animated Solution for Mathematics - Complex Numbers: 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

Visualizing the Argand Plane

Identifying Fixed Points

Geometric Meaning

The Triangle Inequality

Condition for Minimum

Equating to the Given Minimum

Distance Formula Setup

Applying the Magnitude Formula

Squaring Both Sides

Expanding the Squares

Isolating $p^{2}$

Solving for $p$

Final Answer Selection

The Sigma Insight: Geometrical Applications of Complex Numbers

Analyzing the Setup

Translating Algebra into Geometry

The Power of the Triangle Inequality

The Engine Room

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Animated Solution for Mathematics - Complex Numbers: .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