The triangle whose vertices are $A\left(3,-1,2\right),B\left(\lambda ,-1,1\right),C\left(1,1,-2\right)$, is a right-angled triangle. Then the sum of all possible integral values of $\lambda$ is

detailed solution

Correct option is D

Direction ratios of  AB¯,BC¯ ,CA¯are respectively are 3−λ,0,1,λ−1,−2,3,−2,2,−4. In right angle triangle, right angle may occur at any Bvertex   AB¯,BC¯ If the right angle is at the vertex then the dot product of  is zero. 3−λλ−1+3=03λ−3−λ2+λ+3=0λ2−4λ=0λλ−4=0            It givesλ=0,4If the right angle is at the vertex C, then the dot product of   AC¯,BC¯  is zero −23−λ−4−12=02λ−2+16=02λ+14=0λ=−7            It gives  λ=−7 If the right angle is at the vertex Athen the dot product of   AC¯,AB¯  is zero       −23−λ+20−41=0−6+2λ−4=02λ=10λ=5      It gives λ=5Therefore, sum of all the possible values of  λ is   2