Question

The locus of the point of intersection of the two lines \[ x\sqrt{3} - y = k\sqrt{3} \quad \text{and} \quad \sqrt{3}x + ky = \sqrt{3}, \; k \in \mathbb{R}, \] describes

• A. a parabola
• B. a hyperbola
• C. an ellipse
• D. a pair of lines

Hint:

Equations containing an \(xy\) term generally represent a hyperbola (unless reducible).

Answer 1

The Correct Option is B

Step 1: Write the two given equations.
\[ x\sqrt{3} - y = k\sqrt{3} \quad \text{(1)} \] \[ \sqrt{3}x + ky = \sqrt{3} \quad \text{(2)} \]
Step 2: Eliminate the parameter \(k\).
From equation (1): \[ k = \frac{x\sqrt{3}-y}{\sqrt{3}} \]
Step 3: Substitute \(k\) in equation (2).
\[ \sqrt{3}x + y\left(\frac{x\sqrt{3}-y}{\sqrt{3}}\right) = \sqrt{3} \]
Step 4: Simplify the equation.
Multiplying throughout by \(\sqrt{3}\): \[ 3x + y(x\sqrt{3}-y) = 3 \] \[ 3x + \sqrt{3}xy - y^2 = 3 \]
Step 5: Identify the locus.
The obtained equation contains the product term \(xy\), which represents a second–degree curve of hyperbolic nature.
Hence, the locus is a hyperbola.