Three particles start from the origin at the same time, one with a velocity `v_(1)` along the x-axis, second along the y-axis with a velocity `v_(2)` and third particle moves along the line x = y. The velocity of third particle, so that three may always lie on the same line is:

Options

A. `( v_1v_2)/( v_1+v_2) `
B. ` (sqrt(2)v_1v_2)/(v_1+v_2)`
C. ` (sqrt( 3)v_1v_2)/(v_1+v_2) `
D. zero

Explanation

To solve the problem, we need to find the velocity of the third particle moving along the line \( y = x \) such that all three particles remain collinear. Let's break down the solution step by step.

### Step 1: Understand the Motion of Each Particle
- The first particle moves along the x-axis with a velocity \( v_1 \). After time \( t \), its position will be \( (v_1 t, 0) \).
- The second particle moves along the y-axis with a velocity \( v_2 \). After time \( t \), its position will be \( (0, v_2 t) \).
- The third particle moves along the line \( y = x \) with an unknown velocity \( v_3 \). After time \( t \), its position will be \( (x, x) \).

### Step 2: Find the Equation of the Line Connecting the First Two Particles
The line connecting the first two particles can be determined using the two points:
- Point A: \( (v_1 t, 0) \)
- Point B: \( (0, v_2 t) \)

The slope \( m \) of the line connecting these two points is given by:
\[ m = \frac{v_2 t - 0}{0 - v_1 t} = -\frac{v_2}{v_1} \]

Using the point-slope form of the line equation, we can write the equation of the line as:
\[ y - 0 = -\frac{v_2}{v_1}(x - v_1 t) \]
Simplifying this, we get:
\[ y = -\frac{v_2}{v_1}x + v_2 t \]

### Step 3: Find the Condition for Collinearity
For the third particle to be collinear with the first two particles, its position \( (x, x) \) must satisfy the line equation derived above. Therefore, substituting \( y = x \) into the line equation gives:
\[ x = -\frac{v_2}{v_1}x + v_2 t \]
Rearranging this equation:
\[ x + \frac{v_2}{v_1}x = v_2 t \]
\[ x \left(1 + \frac{v_2}{v_1}\right) = v_2 t \]
\[ x = \frac{v_2 t}{1 + \frac{v_2}{v_1}} = \frac{v_2 v_1 t}{v_1 + v_2} \]

### Step 4: Calculate the Distance R
The distance \( R \) traveled by the third particle in time \( t \) is given by:
\[ R = \sqrt{x^2 + x^2} = \sqrt{2x^2} = x\sqrt{2} \]
Substituting the value of \( x \):
\[ R = \frac{v_2 v_1 t}{v_1 + v_2} \sqrt{2} \]

### Step 5: Relate R to the Velocity of the Third Particle
The distance \( R \) can also be expressed in terms of the velocity \( v_3 \) of the third particle:
\[ R = v_3 t \]
Equating the two expressions for \( R \):
\[ v_3 t = \frac{v_2 v_1 t \sqrt{2}}{v_1 + v_2} \]
Dividing both sides by \( t \) (assuming \( t \neq 0 \)):
\[ v_3 = \frac{v_2 v_1 \sqrt{2}}{v_1 + v_2} \]

### Final Answer
Thus, the velocity of the third particle is:
\[ v_3 = \frac{\sqrt{2} v_1 v_2}{v_1 + v_2} \]