Question

Let $ABC$ be an equilateral triangle. If the co-ordinates of $A$ are $(1,2)$ and co-ordinate of $B$ are $(2,-1)$, then

• A. $C$ cannot lie in the first quadrant
• B. $C$ cannot lie in the second quadrant
• C. $C$ is the origin
• D. $C$ cannot lie in the third quadrant

Answer 1

Answer: (B)

$C$ cannot lie in the second quadrant

${(x-1)}^2+{(y-2)}^2={(x-2)}^2+{(y+1)}^2\Rightarrow 2x-6y=0\Rightarrow x=3y$
${(1-2)}^2+{(2+1)}^2={(x-1)}^2+{(y-2)}^2$
$1+9=x^2+1-2x+y^2+4-4y\Rightarrow x^2+y^2-2x-4y-5=0$
$9y^2+y^2-\left(3y\right)-4y-5=0$
$2y^2-2y-1=0\Rightarrow y=\frac{2\pm \sqrt{4+8}}{4}=\frac{1\pm 2\sqrt{3}}{2}\Rightarrow y=1\pm \sqrt{3}$
$\therefore C(3+3\sqrt{3},1+\sqrt{3})or(3-3\sqrt{3},1-\sqrt{3})$