If m is a variable the locus of the point of intersection of the lines `x/3-y/2=m` and `x/3+y/2=1/m` is

Options

A. a parabola
B. an ellipse
C. a hyperbola
D. Straight line

Answer

To find the locus of the point of intersection of the lines given by the equations \( \frac{x}{3} - \frac{y}{2} = m \) and \( \frac{x}{3} + \frac{y}{2} = \frac{1}{m} \), we will follow these steps:

### Step 1: Write the equations
We have two equations:
1. \( \frac{x}{3} - \frac{y}{2} = m \) (Equation 1)
2. \( \frac{x}{3} + \frac{y}{2} = \frac{1}{m} \) (Equation 2)

### Step 2: Eliminate \( m \)
To eliminate \( m \), we can multiply both equations together:
\[ \left(\frac{x}{3} - \frac{y}{2}\right) \left(\frac{x}{3} + \frac{y}{2}\right) = m \cdot \frac{1}{m} \]
This simplifies to:
\[ \left(\frac{x}{3}\right)^2 - \left(\frac{y}{2}\right)^2 = 1 \]

### Step 3: Simplify the equation
Expanding the left-hand side:
\[ \frac{x^2}{9} - \frac{y^2}{4} = 1 \]

### Step 4: Rearranging the equation
Rearranging gives us the standard form of the hyperbola:
\[ \frac{x^2}{3^2} - \frac{y^2}{2^2} = 1 \]

### Conclusion
The locus of the point of intersection of the given lines is a hyperbola represented by the equation:
\[ \frac{x^2}{9} - \frac{y^2}{4} = 1 \]