If three points (0,0),(3, 3) and (3, ) form an equilateral triangle, then =

If three points (0,0),(3, 3) and (3, ) form an equilateral triang…

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MCQ

If three points $(0,0),(3, \sqrt{3})$ and $(3, \lambda)$ form an equilateral triangle, then $\lambda=$

A
$-4$
✓
None of these
C
$-3$
D
$2$

✓

Answer

Correct option: B.

None of these

Explanation:
Let the points $(0,0),(3, \sqrt{3})$ and $(3, \lambda)$ from an equilateral triangle $AB = BC = CA$
$\Rightarrow AB ^2= BC ^2= CA ^2$
Now $, AB ^2=\left( x _2- x _1\right)^2+\left( y _2- y _1\right)^2$
$=(3-0)^2+(\sqrt{3}-0)^2$
$=(3)^2+(\sqrt{3})^2$
$=9+3=12$
$BC ^2=(3-3)^2+(\lambda-\sqrt{3})^2$
$=(0)^2+(\lambda-\sqrt{3})^2=(\lambda-3)^2$
and $ CA ^2=(0-3)^2+(0-\lambda)^2$
$=(-3)^2+(-\lambda)^2$
$=9+\lambda^2$
$A B^2=C A^2 $
$\Rightarrow 12=9+\lambda^2$
$\Rightarrow \lambda^2=12-9=3$
$\therefore \lambda= \pm \sqrt{3}$

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