Question:
If three points $(0,0),(3, \sqrt{3})$ and $(3, \lambda)$ form an equilateral triangle, then $\lambda=$
(a) 2
(b) −3
(c) −4
(d) None of these
Solution:
We have an equilateral triangle $\triangle \mathrm{ABC}$ whose co-ordinates are $\mathrm{A}(0,0) ; \mathrm{B}(3, \sqrt{3})$ and $\mathrm{C}(3, \lambda)$.
Since the triangle is equilateral. So,
$\mathrm{AB}^{2}=\mathrm{AC}^{2}$
So,
$(3-0)^{2}+(\sqrt{3}-0)^{2}=(3-0)^{2}+(\lambda-0)^{2}$
Cancel out the common terms from both the sides,
Therefore,
$\lambda=\sqrt{3}$
So, the answer is (d)