Question

The locus of a variable point which forms a triangle of fixed area with two fixed points is:

• A. a circle.
• B. a circle with fixed points as ends of a diameter.
• C. a pair of non-parallel lines.
• D. a pair of parallel lines.

Hint:

When a variable point forms a triangle of fixed area with two fixed points, its locus is always a pair of parallel lines. This follows from the area determinant equation.

Answer 1

The Correct Option is D

Step 1: Understanding the Geometric Condition
Let the two fixed points be \( A(x_1, y_1) \) and \( B(x_2, y_2) \), and let the variable point be \( P(x, y) \). The area of the triangle formed by these three points is given by the determinant formula: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y) + x_2(y - y_1) + x(y_1 - y_2) \right|. \] For a fixed area \( k \), we set: \[ \frac{1}{2} \left| x_1(y_2 - y) + x_2(y - y_1) + x(y_1 - y_2) \right| = k. \] Step 2: Equation of the Locus Rearranging, \[ \left| x_1(y_2 - y) + x_2(y - y_1) + x(y_1 - y_2) \right| = 2k. \] This represents two straight-line equations, given by: \[ x_1(y_2 - y) + x_2(y - y_1) + x(y_1 - y_2) = \pm 2k. \] Since these are linear equations in \( x \) and \( y \), the locus is a pair of parallel lines.
Step 3: Conclusion
Thus, the correct answer is that the locus of the variable point is a pair of parallel lines.