Question

If the incentre of the triangle formed by lines $$ x - 2 = 0, \quad x + y - 1 = 0, \quad x - y + 3 = 0 $$ is $ (\alpha, \beta) $, then find $ \beta $.

• A. 2
• B. \( \sqrt{2} + 1 \)
• C. \( \frac{2\sqrt{2} - 1}{\sqrt{2} + 1} \)
• D. 4

Hint:

Incentre is weighted average of vertices weighted by side lengths.

Answer 1

The Correct Option is A

1. Find the vertices by solving intersection of lines. 2. Calculate lengths of sides using distance formula. 3. Use formula for incentre coordinates: \[ x = \frac{a x_1 + b x_2 + c x_3}{a + b + c}, \quad y = \frac{a y_1 + b y_2 + c y_3}{a + b + c} \] where \( a, b, c \) are lengths of sides opposite to vertices \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \). 4. After calculation, \( \beta = 2 \).