If chords of the hyperbola $3x^{2}-2y^{2}+4x-6y=0$ are parallel to $y=2x,$ then the locus of the mid points of the chords will be :

Question

Accepted Answer

To determine the locus of the midpoints of chords of the hyperbola $3x^2 - 2y^2 + 4x - 6y = 0$ that are parallel to the line $y = 2x$, we need to follow a systematic approach.

### Explanation:

1. **Equation of Chords Parallel to a Given Line:**
   - Chords parallel to the line $y = 2x$ will have the same slope, i.e., slope $m = 2$.
   - The equation of a line with slope 2 can be written as $y = 2x + c$, where $c$ is a constant.

2. **Midpoint Formula:**
   - For a chord of a conic section, the midpoint $(h, k)$ of the chord can be found using the equation of the conic and the line representing the chord.
   - The midpoint $(h, k)$ of the chord $y = 2x + c$ on the hyperbola can be found using the concept of the chord of contact or by substituting the line equation into the hyperbola and simplifying.

3. **Substitute Line Equation into Hyperbola:**
   - Substitute $y = 2x + c$ into the hyperbola equation:
     $$
     3x^2 - 2(2x + c)^2 + 4x - 6(2x + c) = 0
     $$
   - Simplify the equation:
     $$
     3x^2 - 8x^2 - 8xc - 2c^2 + 4x - 12x - 6c = 0
     $$
     $$
     -5x^2 - 8xc - 2c^2 - 8x - 6c = 0
     $$

4. **Finding the Locus of Midpoints:**
   - The midpoint $(h, k)$ of the chord can be expressed in terms of $c$.
   - By eliminating $c$ from the equations derived from the substitution, we find the relationship between $h$ and $k$.

5. **Derive the Locus Equation:**
   - After simplification, the locus of the midpoints $(h, k)$ is found to satisfy the linear equation:
     $$
     3h - 4k = 4
     $$

### Conclusion:

- **Correct Option:**
  - The equation $3x - 4y = 4$ matches the derived equation for the locus of the midpoints of the chords parallel to $y = 2x$. Therefore, Option 1 is correct.

- **Why Other Options are Incorrect:**
  - **Option 2:** $3y - 4x = 4$ is incorrect because it does not match the derived equation for the locus.
  - **Option 3:** $4x - 4y = 3$ is incorrect as it does not represent the correct relationship between $h$ and $k$.
  - **Option 4:** $3x - 4y = 2$ is incorrect because the constant term does not match the derived equation.

Thus, the correct answer is indeed Option 1: $3x - 4y = 4$.