Complex Numbers and Quadratic Equations · Question 59

Let O be the origin, the point A be z1 = 3 + 2 2i, the point B(z2) be such that 3|z2| = |z1| and (z2) = (z1) + 6. Then

1. area of triangle ABO is 114
2. area of triangle ABO is 11 3
3. ABO is a scalene triangle
4. ABO is an obtuse angled isosceles triangle

Answer

(4) ABO is an obtuse angled isosceles triangle

Solution

OA = |z1| = 3 + 8 = 11

OB = |z2| = |z1| 3 = 113

AOB = (z2) - (z1) = 6

Cosine rule: AB2 = OA2 + OB2 - 2(OA)(OB) 6

= 11 + 113 - 2 11 11 3 32

= 11 + 113 - 11 = 113

AB = 113 = OB the triangle is isosceles.

OA2 = 11 > OB2 + AB2 = 223 OBA is obtuse.

Area = 12(OA)(OB) 6 = 12 11 11 3 12 = 114 3

So options (1) and (2) are wrong, and ABO is not scalene.

ABO is an obtuse angled isosceles triangle.

Answer: (4)