Two particle are moing along `X` and `Y` axes towards the origin with constant speeds `u` and `v` respectively. At time `t=0`, their repective distance from the origin are `x` and `y`. The time instant at which the particles will be closest to each other is

Options

A. `sqrt(x^(2)+y^(2))/sqrt(u^(2)+v^(2))`
B. `(vx+uy)/(u^(2)+v^(2))`
C. `(ux+vy)/(u^(2)+v^(2))`
D. `(2sqrt(x^(2)+y^(2)))/(u+v)`

Explanation

To find the time instant at which two particles moving towards the origin along the X and Y axes will be closest to each other, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Motion**:
- Let particle A be moving along the X-axis with speed \( u \) and starting distance \( x \) from the origin.
- Let particle B be moving along the Y-axis with speed \( v \) and starting distance \( y \) from the origin.

2. **Position of Particles Over Time**:
- The position of particle A at time \( t \) will be \( x_A(t) = x - ut \).
- The position of particle B at time \( t \) will be \( y_B(t) = y - vt \).

3. **Distance Between the Two Particles**:
- The distance \( D \) between the two particles at time \( t \) can be expressed using the Pythagorean theorem:
\[ D(t) = \sqrt{(x - ut)^2 + (y - vt)^2} \]

4. **Finding the Minimum Distance**:
- To find the time when the distance \( D(t) \) is minimized, we can minimize \( D^2(t) \) instead (since the square root function is monotonically increasing):
\[ D^2(t) = (x - ut)^2 + (y - vt)^2 \]

5. **Differentiating with Respect to Time**:
- Differentiate \( D^2(t) \) with respect to \( t \):
\[ \frac{d}{dt} D^2(t) = 2(x - ut)(-u) + 2(y - vt)(-v) \]
- Setting the derivative to zero for minimization gives:
\[ -2u(x - ut) - 2v(y - vt) = 0 \]
- Simplifying this, we get:
\[ u(x - ut) + v(y - vt) = 0 \]

6. **Solving for Time \( t \)**:
- Rearranging the equation:
\[ ux + vy = u^2t + v^2t \]
- Factor out \( t \):
\[ t(u^2 + v^2) = ux + vy \]
- Thus, the time \( t \) when the particles are closest is:
\[ t = \frac{ux + vy}{u^2 + v^2} \]

### Final Answer:
The time instant at which the particles will be closest to each other is:
\[ t = \frac{ux + vy}{u^2 + v^2} \]