The locus of the midpoints of chords of the hyperbola `3x^(2)-2y^(2)+4x-6=0` which are parallel to y= 2x is

Options

A. 3x-4y-4=0
B. 3x-4y+2=0
C. 2x-y=4
D. x+2y=3

Answer

To find the locus of the midpoints of chords of the hyperbola given by the equation \(3x^2 - 2y^2 + 4x - 6 = 0\) that are parallel to the line \(y = 2x\), we can follow these steps:

### Step 1: Rewrite the Hyperbola Equation
First, we need to rewrite the hyperbola equation in standard form. We start with:
\[ 3x^2 - 2y^2 + 4x - 6 = 0 \]
We can rearrange this equation:
\[ 3x^2 + 4x - 2y^2 - 6 = 0 \]

### Step 2: Identify the Midpoint
Let the midpoint of the chord be \((h, k)\). The equation of the chord with midpoint \((h, k)\) can be expressed using the formula:
\[ T = S_1 \]
where \(T\) is the equation obtained by replacing \(x\) with \(h\) and \(y\) with \(k\) in the hyperbola equation, and \(S_1\) is the value of the hyperbola at the point \((h, k)\).

### Step 3: Substitute in the Hyperbola Equation
Substituting \(x = h\) and \(y = k\) into the hyperbola equation gives:
\[ 3h^2 - 2k^2 + 4h - 6 = 0 \]
This is our \(S_1\).

### Step 4: Write the Equation of the Chord
The equation of the chord can be expressed as:
\[ 3hx - 2ky + 2h + 4h - 6 = 3h^2 - 2k^2 + 4h \]
Simplifying this gives:
\[ 3hx - 2ky + 2h - 3h^2 + 2k^2 = 0 \]

### Step 5: Find the Slope of the Chord
The slope of the chord is given by:
\[ \text{slope} = -\frac{A}{B} = -\frac{3h + 2}{-2k} = \frac{3h + 2}{2k} \]

### Step 6: Set the Slope Equal to the Given Line
Since the chord is parallel to the line \(y = 2x\), the slope of the chord must equal 2:
\[ \frac{3h + 2}{2k} = 2 \]

### Step 7: Solve for \(h\) and \(k\)
Cross-multiplying gives:
\[ 3h + 2 = 4k \]
Rearranging this gives:
\[ 3h - 4k + 2 = 0 \]

### Step 8: Replace \(h\) and \(k\) with \(x\) and \(y\)
To find the locus, we replace \(h\) and \(k\) with \(x\) and \(y\):
\[ 3x - 4y + 2 = 0 \]
or equivalently:
\[ 3x - 4y = -2 \]

### Final Result
Thus, the locus of the midpoints of the chords of the hyperbola that are parallel to the line \(y = 2x\) is given by:
\[ 3x - 4y = -2 \]