Let ABCD be a quadrilateral with AB = a , AD = b and AC =3 a +2 ~b . If its area is times the area of the parallelogram with AB , AD as adjacent sides, then the value of is equal to

Step-by-step solution

Position vectors: Let A = 0 , AB = a , AD = b , and AC = 3 a + 2 b . The area of the parallelogram with adjacent sides AB and AD is P = | a b | . Quadrilateral ABCD is divided by diagonal AC into two triangles. The area of ABC is 1 2 | a (3 a + 2 b )| = 1 2 |2( a b )| = | a b | . The area of ADC is 1 2 | b (3 a + 2 b )| = 1 2 |-3( a b )| = 3 2 | a b | . Total area is | a b | + 3 2 | a b | = 5 2 | a b | . Alternatively, using diagonals AC = 3 a + 2 b and BD = b - a , their cross product is 5( a b ) , giving area 5 2