The locus of point of intersection of the lines `x/a-y/b=m` and `x/a+y/b=1/m` (i) a circle (ii) an ellipse (iii) a hyperbola (iv) a parabola

Options

A. a circle
B. an ellipse
C. a hyperbola
D. a parabola

Answer

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

### Step 1: Write down the equations
The two equations are:
1. \( \frac{x}{a} - \frac{y}{b} = m \) (Equation 1)
2. \( \frac{x}{a} + \frac{y}{b} = \frac{1}{m} \) (Equation 2)

### Step 2: Solve for \( x \) and \( y \)
To find the point of intersection, we can add and subtract the two equations.

Adding the two equations:
\[ \left(\frac{x}{a} - \frac{y}{b}\right) + \left(\frac{x}{a} + \frac{y}{b}\right) = m + \frac{1}{m} \]
This simplifies to:
\[ \frac{2x}{a} = m + \frac{1}{m} \]
Thus, we can express \( x \) as:
\[ x = \frac{a}{2} \left(m + \frac{1}{m}\right) \]

Now, subtracting the second equation from the first:
\[ \left(\frac{x}{a} - \frac{y}{b}\right) - \left(\frac{x}{a} + \frac{y}{b}\right) = m - \frac{1}{m} \]
This simplifies to:
\[ -\frac{2y}{b} = m - \frac{1}{m} \]
Thus, we can express \( y \) as:
\[ y = -\frac{b}{2} \left(m - \frac{1}{m}\right) \]

### Step 3: Eliminate \( m \)
Now, we have expressions for \( x \) and \( y \) in terms of \( m \):
\[ x = \frac{a}{2} \left(m + \frac{1}{m}\right) \]
\[ y = -\frac{b}{2} \left(m - \frac{1}{m}\right) \]

To eliminate \( m \), we can express \( m \) in terms of \( x \) and \( y \).

From the expression for \( x \):
\[ 2x = a \left(m + \frac{1}{m}\right) \]
Multiplying both sides by \( m \):
\[ 2xm = am + a \]
Rearranging gives:
\[ 2xm - am = a \]
\[ m(2x - a) = a \]
Thus,
\[ m = \frac{a}{2x - a} \]

Now substituting this value of \( m \) into the expression for \( y \):
\[ y = -\frac{b}{2} \left(\frac{a}{2x - a} - \frac{2x - a}{a}\right) \]

### Step 4: Simplify and find the locus
After substituting and simplifying, we will arrive at an equation relating \( x \) and \( y \). The final equation will be of the form:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
This is the standard form of a hyperbola.

### Conclusion
Thus, the locus of the point of intersection of the lines is a hyperbola.