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

The area of the parallelogram with adjacent sides AB and AD is given by: ₁ = | AB AD | = | a b | The area of the quadrilateral ABCD can be calculated using its diagonals AC and BD : ₂ = 1 2 | AC BD | The diagonal vector BD is: BD = AD - AB = b - a Taking the cross product of the diagonals: AC BD = (3 a + 2 b ) ( b - a ) AC BD = 3( a b ) - 3( a a ) + 2( b b ) - 2( b a ) Since a a = 0 , b b = 0 , and b a = -( a b ) : AC BD = 3( a b ) - 0 + 0 + 2( a b ) = 5( a b ) Substituting this back into the area formula for the q