A triangle `DeltaABC` has sides of lengths `|z_1|,|z_2| and |z_3|` such that `|z_1| (|z_1 |-|z_2|)+ |z_2| (|z_2 |-|z_3|)+|z_3| (|z_3 |-|z_1|)=0` Then which of the following can be a possible pair of values of `(|z_1|R)` (R = circumradius of `DeltaABC` ) ?

Options

A. `(1,sqrt2)`
B. `(sqrt2,1)`
C. `(1,sqrt3)`
D. `(sqrt3,1)`

Answer

To solve the problem, we start with the given equation for the triangle \( \Delta ABC \):

\[ |z_1| (|z_1| - |z_2|) + |z_2| (|z_2| - |z_3|) + |z_3| (|z_3| - |z_1|) = 0 \]

### Step 1: Expand the Equation
We will expand the equation:

\[ |z_1|^2 - |z_1||z_2| + |z_2|^2 - |z_2||z_3| + |z_3|^2 - |z_3||z_1| = 0 \]

Rearranging gives us:

\[ |z_1|^2 + |z_2|^2 + |z_3|^2 = |z_1||z_2| + |z_2||z_3| + |z_3||z_1| \]

### Step 2: Recognize the Condition for Equality
The equality \( |z_1|^2 + |z_2|^2 + |z_3|^2 = |z_1||z_2| + |z_2||z_3| + |z_3||z_1| \) holds true if and only if \( |z_1| = |z_2| = |z_3| \). This means that the triangle is equilateral.

### Step 3: Relate Side Lengths to Circumradius
For an equilateral triangle, the relationship between the side length \( a \) and the circumradius \( R \) is given by:

\[ a = \sqrt{3} R \]

### Step 4: Find the Pair \( (|z_1|, R) \)
If we let \( |z_1| = a \), then we have:

\[ |z_1| = a \quad \text{and} \quad R = \frac{a}{\sqrt{3}} \]

Thus, the product \( |z_1| R \) becomes:

\[ |z_1| R = a \cdot \frac{a}{\sqrt{3}} = \frac{a^2}{\sqrt{3}} \]

### Step 5: Possible Values of \( (|z_1| R) \)
Now, we can express \( |z_1| R \) in terms of \( a \):

If we take \( a = \sqrt{3} \), then:

\[ |z_1| R = \frac{(\sqrt{3})^2}{\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3} \]

Thus, the possible pair of values of \( (|z_1| R) \) can be \( (\sqrt{3}, 1) \).

### Conclusion
The possible pair of values of \( (|z_1| R) \) is \( \sqrt{3} \) and \( 1 \), which corresponds to the option that states this pair.