If the line l 1 : 3 y - 2 x = 3 is the angular bisector of the lines l 2 : x - y + 1 = 0 and l 3 : x + y + 17 = 0 , then 2 + 2 is equal to ______ . [2023]

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Solution

Q.
If the line $l_{1} : 3 y - 2 x = 3$ is the angular bisector of the lines $l_{2} : x - y + 1 = 0 and l_{3} : α x + β y + 17 = 0,$ then $α^{2} + β^{2} - α - β$ is equal to ______ . [2023]

Ans.
(348)

$Point of intersection of l_{1} : 3 y - 2 x = 3 and l_{2} : x - y + 1 = 0 is P (0, 1),$

$which lies on l_{3} : α x + β y + 17 = 0, β = - 17$

$Consider a random point Q (- 1, 0) on l_{2} : x - y + 1 = 0$

$Image of point Q about l_{1} : 2 x - 3 y + 3 = 0 is Q' (\frac{- 17}{13}, \frac{6}{13}) .$

$which can be calculated by the formula,$

$\frac{x - (- 1)}{2} = \frac{y - 0}{- 3} = \frac{- 2 (- 2 + 3)}{13}$

$Now, Q' lies on l_{3} : α x + β y + 17 = 0$

$∴$ $α = 7$

$Now, α^{2} + β^{2} - α - β = 348$

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