A particle moves in x-y plane under action of a path dependent force `F=yhati+xhatj.` The work done by the force as it moves in x-y plane can be evaluated by solving the integral `intvecF.vecdr,`where `vec(dr)=dxhati+dyhatj.` The position coordinates x and y will vary according to some constraint determined by teh path followed by the particle. For example, if particle moves along a straight line from one position to other, then an possible rarticle. For example, if particle moves along a straight line from one position to other, then a possible relation is `y=mx+c.` NOw. try to solve the following questions.
1. The particel moves along a straight line from origin to (a ,a) . The work done by the force on the particle is
`(1) a^(2)" "(2)2s" "(3)(a^(2))/(2)" "(4)`Zero
2. The particle moves from `(0,0)` to (a,0) and then from, `(a,0)` to `(a,a),` in straight line paths. The work done by the force on the particle is
`(1)a^(2)" "(2)2a" "(3)(a^(2))/(2)" "(4)` Zero
3. The differential work by the given force over an elementary displacement is given as `dW=vecF.vec(dr)=ydx+xdy`
This can be expressed as `dW=d(xy).` From this, it can be inferred that the work done by the force
(1) Depends only on initial and final values of x and y
(2) Depends on initial and final values and on the path followed
(3) Is zero for any values of x and y
(4) Is zero when object returns to original position after following any path

Explanation

To solve the given questions step by step, we will analyze each part of the problem systematically.

### Question 1:
**The particle moves along a straight line from origin to (a, a). The work done by the force on the particle is:**

1. **Identify the path:** The particle moves from (0, 0) to (a, a). The equation of the line is \(y = x\).

2. **Express \(dy\) in terms of \(dx\):** Since \(y = x\), we have \(dy = dx\).
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