Question

Let $A \left( z _1\right), B \left( z _2\right)$ and $C \left( z _3\right)$ be the vertices of $\triangle ABC$ such that $\left| z _1- z _2\right|=3, z _1{ }^2= z _2 z _3, z _2{ }^2= z _1 z _3$ and $z_1 z_2 z_3 \neq 0$. Then which of the following is/are correct?

• A. Area of $\triangle ABC =\frac{9 \sqrt{3}}{4} $
• B. Circumradius of $\triangle ABC =\sqrt{3}$
• C. Area of $\triangle ABC =3$
• D. $\triangle ABC$ is an isosceles triangle

Answer 1

Answer: (A), (C), (D)

$ z _1{ }^2= z _2 z _3 \text { and } z _2{ }^2= z _1 z _3 $
$\Rightarrow z _3{ }^3= z _1 z _2 $
$\Rightarrow z _1{ }^2+ z _2{ }^2+ z _3{ }^2= z _1 z _2+ z _2 z _3+ z _3 z _1$
$\Rightarrow \Delta ABC \text { is an equilateral triangle. }$