Question

Find the locus of the incenter of the triangle formed by: \[ xy - 4x - 4y + 16 = 0,\quad x + y = 5 \]

• A. \( x - y = 0 \)
• B. \( x + y = 0 \)
• C. \( x - 2y = 0 \)
• D. \( 2x - y = 0 \)

Hint:

When triangle sides are symmetric about \( x = y \), and all angles are equal (like 45°), incenter lies on the angle bisector — here, the line \( x = y \).

Answer 1

The Correct Option is A

The first equation represents a pair of lines: \[ xy - 4x - 4y + 16 = 0 \Rightarrow (x - 4)(y - 4) = 0 \Rightarrow x = 4,\ y = 4 \] So, the triangle is formed by lines: - \( x = 4 \) - \( y = 4 \) - \( x + y = 5 \) These three lines form a triangle in the first quadrant. To find the locus of the incenter as the triangle moves (with parameter), observe symmetry. By geometry and from solving incenter coordinates for triangle formed by: - Vertical, horizontal, and slanted line — the incenter lies along \( x = y \)