If the vertices of a triangle are 1,2,3,3,4,−1,5,−4,2and the area of triangle is λ , then the number of positive integral divisors of  λ26+λ2+253 is

The area of the triangle formed by the vertices A, B, C is 12AB¯×AC¯Here,             AB¯=OB¯−OA¯=3i+4j−k−i−2j−3k=2i+2j−4kAC=OC−OA=5i−4j+2k−i−2j−3k=4i−6j−kHence, the area of the triangle is 12AB¯×AC¯=12ijk22−44−6−1=12i−2−24−j−2+16+k−12−8=12−26i−14j−20k=318  Consider ,        λ26+λ2+253=3186+318+253=53+3433=60=22⋅3⋅5    The number of integral divisors of the above number is  2+1⋅1+1⋅1+1=12